| L(s) = 1 | + 32·2-s − 108·3-s + 640·4-s − 500·5-s − 3.45e3·6-s + 425·7-s + 1.02e4·8-s + 7.29e3·9-s − 1.60e4·10-s + 5.83e3·11-s − 6.91e4·12-s + 8.78e3·13-s + 1.36e4·14-s + 5.40e4·15-s + 1.43e5·16-s + 6.37e3·17-s + 2.33e5·18-s − 2.39e4·19-s − 3.20e5·20-s − 4.59e4·21-s + 1.86e5·22-s − 5.68e4·23-s − 1.10e6·24-s + 1.56e5·25-s + 2.81e5·26-s − 3.93e5·27-s + 2.72e5·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 2.30·3-s + 5·4-s − 1.78·5-s − 6.53·6-s + 0.468·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s + 1.32·11-s − 11.5·12-s + 1.10·13-s + 1.32·14-s + 4.13·15-s + 35/4·16-s + 0.314·17-s + 9.42·18-s − 0.801·19-s − 8.94·20-s − 1.08·21-s + 3.73·22-s − 0.975·23-s − 16.3·24-s + 2·25-s + 3.13·26-s − 3.84·27-s + 2.34·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(33.74662302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(33.74662302\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 5 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 425 T + 104614 p T^{2} - 258353405 T^{3} - 122056990070 T^{4} - 258353405 p^{7} T^{5} + 104614 p^{15} T^{6} - 425 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 5835 T + 279476 p^{2} T^{2} + 40245444333 T^{3} - 140733403350810 T^{4} + 40245444333 p^{7} T^{5} + 279476 p^{16} T^{6} - 5835 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 375 p T + 805950380 T^{2} - 1910517768369 T^{3} + 418959064006514742 T^{4} - 1910517768369 p^{7} T^{5} + 805950380 p^{14} T^{6} - 375 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 23956 T + 2102231728 T^{2} + 71201050326196 T^{3} + 2189393255471331694 T^{4} + 71201050326196 p^{7} T^{5} + 2102231728 p^{14} T^{6} + 23956 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 56895 T + 7200687698 T^{2} + 409551267109995 T^{3} + 36877892066475192858 T^{4} + 409551267109995 p^{7} T^{5} + 7200687698 p^{14} T^{6} + 56895 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 14922 T + 46165285904 T^{2} - 674478468558126 T^{3} + \)\(11\!\cdots\!82\)\( T^{4} - 674478468558126 p^{7} T^{5} + 46165285904 p^{14} T^{6} - 14922 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 338626 T + 124038678052 T^{2} + 24413167724807962 T^{3} + \)\(50\!\cdots\!38\)\( T^{4} + 24413167724807962 p^{7} T^{5} + 124038678052 p^{14} T^{6} + 338626 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 692629 T + 504310886830 T^{2} + 196163945172665791 T^{3} + \)\(76\!\cdots\!02\)\( T^{4} + 196163945172665791 p^{7} T^{5} + 504310886830 p^{14} T^{6} + 692629 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 169989 T + 130257960110 T^{2} + 71727872628139011 T^{3} + \)\(81\!\cdots\!34\)\( T^{4} + 71727872628139011 p^{7} T^{5} + 130257960110 p^{14} T^{6} + 169989 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1068356 T + 1262768554300 T^{2} - 850379655400104356 T^{3} + \)\(53\!\cdots\!90\)\( T^{4} - 850379655400104356 p^{7} T^{5} + 1262768554300 p^{14} T^{6} - 1068356 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 476898 T + 961552534748 T^{2} - 114426802653969402 T^{3} + \)\(43\!\cdots\!10\)\( T^{4} - 114426802653969402 p^{7} T^{5} + 961552534748 p^{14} T^{6} - 476898 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 1826847 T + 3731982452858 T^{2} + 3846462122086241517 T^{3} + \)\(53\!\cdots\!54\)\( T^{4} + 3846462122086241517 p^{7} T^{5} + 3731982452858 p^{14} T^{6} + 1826847 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 318240 T + 1619331456860 T^{2} + 4278515728114501920 T^{3} + \)\(68\!\cdots\!58\)\( T^{4} + 4278515728114501920 p^{7} T^{5} + 1619331456860 p^{14} T^{6} + 318240 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 1151893 T + 6799637930746 T^{2} + 8607301072016472199 T^{3} + \)\(26\!\cdots\!50\)\( T^{4} + 8607301072016472199 p^{7} T^{5} + 6799637930746 p^{14} T^{6} + 1151893 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 3451700 T + 2129124264268 T^{2} - 13983230631180400436 T^{3} + \)\(74\!\cdots\!22\)\( T^{4} - 13983230631180400436 p^{7} T^{5} + 2129124264268 p^{14} T^{6} - 3451700 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 7491075 T + 18839205086564 T^{2} + 42093930862375235505 T^{3} - \)\(30\!\cdots\!10\)\( T^{4} + 42093930862375235505 p^{7} T^{5} + 18839205086564 p^{14} T^{6} - 7491075 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8780378 T + 59424138073888 T^{2} - \)\(28\!\cdots\!34\)\( T^{3} + \)\(10\!\cdots\!34\)\( T^{4} - \)\(28\!\cdots\!34\)\( p^{7} T^{5} + 59424138073888 p^{14} T^{6} - 8780378 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11445371 T + 88641647374324 T^{2} - 6618196340776041809 p T^{3} + \)\(25\!\cdots\!46\)\( T^{4} - 6618196340776041809 p^{8} T^{5} + 88641647374324 p^{14} T^{6} - 11445371 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 23765676 T + 313869168372956 T^{2} - \)\(26\!\cdots\!80\)\( T^{3} + \)\(16\!\cdots\!02\)\( T^{4} - \)\(26\!\cdots\!80\)\( p^{7} T^{5} + 313869168372956 p^{14} T^{6} - 23765676 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 16294497 T + 126650317757918 T^{2} - \)\(30\!\cdots\!99\)\( T^{3} + \)\(52\!\cdots\!62\)\( T^{4} - \)\(30\!\cdots\!99\)\( p^{7} T^{5} + 126650317757918 p^{14} T^{6} - 16294497 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 10066559 T + 235076729883556 T^{2} - \)\(17\!\cdots\!45\)\( T^{3} + \)\(24\!\cdots\!26\)\( T^{4} - \)\(17\!\cdots\!45\)\( p^{7} T^{5} + 235076729883556 p^{14} T^{6} - 10066559 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.76727919731863590157276533827, −6.45103956321887500871684927015, −6.40432140107235981008357808481, −6.15827947639402046688865517048, −6.08029139767610260037722226775, −5.46732004022414869593858625834, −5.32000083491963904817293459556, −5.15071300261981792260185201832, −5.14186076081049694371345070585, −4.46434901282969791711815766393, −4.40714599537620283037445210230, −4.39316080706527206137770817825, −3.89655839114885485673297481447, −3.66358594797488815412292254936, −3.43245517147617162295381955746, −3.39544025226141157475801223215, −3.25938807373541207589354150422, −2.11131458265283436915955214297, −2.04808899303251947551700456065, −1.93977209669169118890312323921, −1.73855107083275179480818088591, −0.991963731885094117360851227286, −0.74276100842823323256807498051, −0.55087065869065764997521162169, −0.50747672952145583734446394028,
0.50747672952145583734446394028, 0.55087065869065764997521162169, 0.74276100842823323256807498051, 0.991963731885094117360851227286, 1.73855107083275179480818088591, 1.93977209669169118890312323921, 2.04808899303251947551700456065, 2.11131458265283436915955214297, 3.25938807373541207589354150422, 3.39544025226141157475801223215, 3.43245517147617162295381955746, 3.66358594797488815412292254936, 3.89655839114885485673297481447, 4.39316080706527206137770817825, 4.40714599537620283037445210230, 4.46434901282969791711815766393, 5.14186076081049694371345070585, 5.15071300261981792260185201832, 5.32000083491963904817293459556, 5.46732004022414869593858625834, 6.08029139767610260037722226775, 6.15827947639402046688865517048, 6.40432140107235981008357808481, 6.45103956321887500871684927015, 6.76727919731863590157276533827
