| L(s) = 1 | + 6·2-s − 64·4-s − 252·5-s − 384·8-s − 1.51e3·10-s + 3.97e3·11-s − 1.17e3·13-s + 5.55e3·16-s − 5.63e4·17-s + 4.17e4·19-s + 1.61e4·20-s + 2.38e4·22-s − 1.31e5·23-s − 1.58e5·25-s − 7.05e3·26-s − 3.40e4·29-s + 4.01e5·31-s + 9.72e4·32-s − 3.38e5·34-s − 5.39e3·37-s + 2.50e5·38-s + 9.67e4·40-s − 4.10e5·41-s + 4.65e4·43-s − 2.54e5·44-s − 7.90e5·46-s − 1.47e6·47-s + ⋯ |
| L(s) = 1 | + 0.530·2-s − 1/2·4-s − 0.901·5-s − 0.265·8-s − 0.478·10-s + 0.899·11-s − 0.148·13-s + 0.338·16-s − 2.78·17-s + 1.39·19-s + 0.450·20-s + 0.477·22-s − 2.25·23-s − 2.02·25-s − 0.0787·26-s − 0.259·29-s + 2.41·31-s + 0.524·32-s − 1.47·34-s − 0.0175·37-s + 0.740·38-s + 0.239·40-s − 0.930·41-s + 0.0892·43-s − 0.449·44-s − 1.19·46-s − 2.06·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 3 p T + 25 p^{2} T^{2} - 75 p^{3} T^{3} + 67 p^{5} T^{4} - 75 p^{10} T^{5} + 25 p^{16} T^{6} - 3 p^{22} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 252 T + 44402 p T^{2} + 1577184 p^{2} T^{3} + 179949007 p^{3} T^{4} + 1577184 p^{9} T^{5} + 44402 p^{15} T^{6} + 252 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 3972 T + 60322210 T^{2} - 213160200216 T^{3} + 1598816883729563 T^{4} - 213160200216 p^{7} T^{5} + 60322210 p^{14} T^{6} - 3972 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 1176 T + 151551116 T^{2} + 19538457960 p T^{3} + 75518676758118 p^{2} T^{4} + 19538457960 p^{8} T^{5} + 151551116 p^{14} T^{6} + 1176 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 56364 T + 2467770938 T^{2} + 71902986280272 T^{3} + 1676500251773703363 T^{4} + 71902986280272 p^{7} T^{5} + 2467770938 p^{14} T^{6} + 56364 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 41748 T + 3744858290 T^{2} - 107414138569080 T^{3} + 5091577232715281931 T^{4} - 107414138569080 p^{7} T^{5} + 3744858290 p^{14} T^{6} - 41748 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 131748 T + 15612629090 T^{2} + 1076115572991720 T^{3} + 75420655452282784779 T^{4} + 1076115572991720 p^{7} T^{5} + 15612629090 p^{14} T^{6} + 131748 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 34056 T + 37434723340 T^{2} - 965899818999528 T^{3} + \)\(63\!\cdots\!58\)\( T^{4} - 965899818999528 p^{7} T^{5} + 37434723340 p^{14} T^{6} + 34056 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 401212 T + 139975157906 T^{2} - 31935313732549752 T^{3} + \)\(61\!\cdots\!59\)\( T^{4} - 31935313732549752 p^{7} T^{5} + 139975157906 p^{14} T^{6} - 401212 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 5396 T + 363127443338 T^{2} + 1825865606665728 T^{3} + \)\(50\!\cdots\!23\)\( T^{4} + 1825865606665728 p^{7} T^{5} + 363127443338 p^{14} T^{6} + 5396 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 410424 T + 514131659548 T^{2} + 263929507132312392 T^{3} + \)\(12\!\cdots\!54\)\( T^{4} + 263929507132312392 p^{7} T^{5} + 514131659548 p^{14} T^{6} + 410424 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 46544 T + 620213285356 T^{2} - 12386906008354640 T^{3} + \)\(22\!\cdots\!42\)\( T^{4} - 12386906008354640 p^{7} T^{5} + 620213285356 p^{14} T^{6} - 46544 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 1470084 T + 2222554120274 T^{2} + 1971963683479379016 T^{3} + \)\(17\!\cdots\!11\)\( T^{4} + 1971963683479379016 p^{7} T^{5} + 2222554120274 p^{14} T^{6} + 1470084 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 642372 T + 3010217764922 T^{2} - 2146829061764892384 T^{3} + \)\(47\!\cdots\!03\)\( T^{4} - 2146829061764892384 p^{7} T^{5} + 3010217764922 p^{14} T^{6} - 642372 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 752220 T + 4552033960274 T^{2} + 1079212006097726760 T^{3} + \)\(11\!\cdots\!15\)\( T^{4} + 1079212006097726760 p^{7} T^{5} + 4552033960274 p^{14} T^{6} + 752220 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1325772 T + 4415898948746 T^{2} + 1495430708559062688 T^{3} + \)\(23\!\cdots\!39\)\( T^{4} + 1495430708559062688 p^{7} T^{5} + 4415898948746 p^{14} T^{6} - 1325772 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 290916 T + 11027966376914 T^{2} - 2788765142144857416 T^{3} + \)\(60\!\cdots\!11\)\( T^{4} - 2788765142144857416 p^{7} T^{5} + 11027966376914 p^{14} T^{6} + 290916 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 3377760 T + 21455628109916 T^{2} + 39866994142106473440 T^{3} + \)\(20\!\cdots\!26\)\( T^{4} + 39866994142106473440 p^{7} T^{5} + 21455628109916 p^{14} T^{6} + 3377760 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6706588 T + 49059611558282 T^{2} - \)\(20\!\cdots\!96\)\( T^{3} + \)\(84\!\cdots\!03\)\( T^{4} - \)\(20\!\cdots\!96\)\( p^{7} T^{5} + 49059611558282 p^{14} T^{6} - 6706588 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 3946244 T + 50171320232098 T^{2} - 80373712829364371720 T^{3} + \)\(10\!\cdots\!79\)\( T^{4} - 80373712829364371720 p^{7} T^{5} + 50171320232098 p^{14} T^{6} - 3946244 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 9542064 T + 123192866313548 T^{2} + \)\(78\!\cdots\!08\)\( T^{3} + \)\(52\!\cdots\!18\)\( T^{4} + \)\(78\!\cdots\!08\)\( p^{7} T^{5} + 123192866313548 p^{14} T^{6} + 9542064 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 16165212 T + 233578730880010 T^{2} + \)\(20\!\cdots\!40\)\( T^{3} + \)\(16\!\cdots\!11\)\( T^{4} + \)\(20\!\cdots\!40\)\( p^{7} T^{5} + 233578730880010 p^{14} T^{6} + 16165212 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1533112 T + 125450995201724 T^{2} - \)\(51\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!22\)\( T^{4} - \)\(51\!\cdots\!40\)\( p^{7} T^{5} + 125450995201724 p^{14} T^{6} - 1533112 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.22370689657905264097921875146, −7.19091181524729618152903241613, −6.64788275573201067224946040617, −6.62875562632456165751669816394, −6.27460295879846032799647471256, −6.25468749296645285363572827868, −5.80065730461907802958066541318, −5.66608611783909450447934626633, −5.35235137062566443976059718250, −4.84411514870933069029948259094, −4.72266067488451628736708667906, −4.51053921331913259046101137557, −4.44011476455477270870109067704, −3.94639904877733401403592233090, −3.77452071905038251038317362389, −3.67789732357287426031219090088, −3.47361930353565199882044928686, −2.87494378882665989512133378805, −2.65323004117306505151028300454, −2.35755624612048212296137938252, −2.02788701713840486262303283147, −1.82792312879737546397840442782, −1.37502009511481150213498949734, −1.07011418148179243069892446338, −0.906275566380818105384172980260, 0, 0, 0, 0,
0.906275566380818105384172980260, 1.07011418148179243069892446338, 1.37502009511481150213498949734, 1.82792312879737546397840442782, 2.02788701713840486262303283147, 2.35755624612048212296137938252, 2.65323004117306505151028300454, 2.87494378882665989512133378805, 3.47361930353565199882044928686, 3.67789732357287426031219090088, 3.77452071905038251038317362389, 3.94639904877733401403592233090, 4.44011476455477270870109067704, 4.51053921331913259046101137557, 4.72266067488451628736708667906, 4.84411514870933069029948259094, 5.35235137062566443976059718250, 5.66608611783909450447934626633, 5.80065730461907802958066541318, 6.25468749296645285363572827868, 6.27460295879846032799647471256, 6.62875562632456165751669816394, 6.64788275573201067224946040617, 7.19091181524729618152903241613, 7.22370689657905264097921875146
