| L(s) = 1 | + 6·3-s − 8·7-s + 9·9-s + 6·11-s − 20·13-s − 76·19-s − 48·21-s − 24·23-s − 54·27-s − 60·29-s − 4·31-s + 36·33-s − 56·37-s − 120·39-s − 90·41-s + 22·43-s − 204·47-s + 54·49-s − 456·57-s − 174·59-s + 44·61-s − 72·63-s − 14·67-s − 144·69-s + 148·73-s − 48·77-s − 160·79-s + ⋯ |
| L(s) = 1 | + 2·3-s − 8/7·7-s + 9-s + 6/11·11-s − 1.53·13-s − 4·19-s − 2.28·21-s − 1.04·23-s − 2·27-s − 2.06·29-s − 0.129·31-s + 1.09·33-s − 1.51·37-s − 3.07·39-s − 2.19·41-s + 0.511·43-s − 4.34·47-s + 1.10·49-s − 8·57-s − 2.94·59-s + 0.721·61-s − 8/7·63-s − 0.208·67-s − 2.08·69-s + 2.02·73-s − 0.623·77-s − 2.02·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.07589931784\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07589931784\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T + 10 T^{2} - 352 T^{3} - 2621 T^{4} - 352 p^{2} T^{5} + 10 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 7 p T^{2} - 390 T^{3} - 8964 T^{4} - 390 p^{2} T^{5} + 7 p^{5} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 20 T + 22 T^{2} + 800 T^{3} + 46723 T^{4} + 800 p^{2} T^{5} + 22 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 790 T^{2} + 320907 T^{4} - 790 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 2 p T + 1023 T^{2} + 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T + 578 T^{2} + 9264 T^{3} - 29277 T^{4} + 9264 p^{2} T^{5} + 578 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 60 T + 2462 T^{2} + 75720 T^{3} + 1894563 T^{4} + 75720 p^{2} T^{5} + 2462 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 1030 T^{2} - 11744 T^{3} + 89299 T^{4} - 11744 p^{2} T^{5} + 1030 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 45 T + 2356 T^{2} + 45 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 22 T - 2375 T^{2} + 18458 T^{3} + 3860164 T^{4} + 18458 p^{2} T^{5} - 2375 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 204 T + 21578 T^{2} + 1572024 T^{3} + 85146003 T^{4} + 1572024 p^{2} T^{5} + 21578 p^{4} T^{6} + 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9700 T^{2} + 38880102 T^{4} - 9700 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 174 T + 19397 T^{2} + 1619070 T^{3} + 109595916 T^{4} + 1619070 p^{2} T^{5} + 19397 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 44 T - 5450 T^{2} + 2464 T^{3} + 33503299 T^{4} + 2464 p^{2} T^{5} - 5450 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T - 191 T^{2} - 120274 T^{3} - 20881196 T^{4} - 120274 p^{2} T^{5} - 191 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9148 T^{2} + 46550598 T^{4} - 9148 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 74 T + 11487 T^{2} - 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 160 T + 7258 T^{2} + 937600 T^{3} + 137709283 T^{4} + 937600 p^{2} T^{5} + 7258 p^{4} T^{6} + 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - 156 T + 15001 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 22468 T^{2} + 231780678 T^{4} - 22468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 8675 T^{2} - 20278 T^{3} - 13241876 T^{4} - 20278 p^{2} T^{5} - 8675 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.93625202827040468967225724699, −6.86306727801139383849454813930, −6.80545397228940261556600705744, −6.27671508179489740543241198405, −6.19813867227101876376535716223, −6.13165309900273483910042483003, −5.77625336985699167514072422030, −5.29299989241751866992758708943, −5.09921455488945899982913128115, −4.97056562165564338275791748654, −4.70402126495135318967294809862, −4.18230861928418070100160937410, −3.94685821346819182096405954922, −3.85096056184315799706683157028, −3.72584242313726120986194453006, −3.39889213775023137044361227076, −2.91963804336873860823203201221, −2.78543258921820133796550094277, −2.72383767258915603517396887618, −2.06111382465376767941664283652, −1.87326815575720668527019014344, −1.76990253087900819138383100948, −1.68694719883067815938096608480, −0.25716528976100095628024961561, −0.083806940619888558118044741468,
0.083806940619888558118044741468, 0.25716528976100095628024961561, 1.68694719883067815938096608480, 1.76990253087900819138383100948, 1.87326815575720668527019014344, 2.06111382465376767941664283652, 2.72383767258915603517396887618, 2.78543258921820133796550094277, 2.91963804336873860823203201221, 3.39889213775023137044361227076, 3.72584242313726120986194453006, 3.85096056184315799706683157028, 3.94685821346819182096405954922, 4.18230861928418070100160937410, 4.70402126495135318967294809862, 4.97056562165564338275791748654, 5.09921455488945899982913128115, 5.29299989241751866992758708943, 5.77625336985699167514072422030, 6.13165309900273483910042483003, 6.19813867227101876376535716223, 6.27671508179489740543241198405, 6.80545397228940261556600705744, 6.86306727801139383849454813930, 6.93625202827040468967225724699
