| L(s) = 1 | + 4·2-s + 4·3-s + 10·4-s − 4·5-s + 16·6-s − 6·7-s + 20·8-s + 10·9-s − 16·10-s − 4·11-s + 40·12-s − 5·13-s − 24·14-s − 16·15-s + 35·16-s − 10·17-s + 40·18-s − 8·19-s − 40·20-s − 24·21-s − 16·22-s − 7·23-s + 80·24-s − 5·25-s − 20·26-s + 20·27-s − 60·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.78·5-s + 6.53·6-s − 2.26·7-s + 7.07·8-s + 10/3·9-s − 5.05·10-s − 1.20·11-s + 11.5·12-s − 1.38·13-s − 6.41·14-s − 4.13·15-s + 35/4·16-s − 2.42·17-s + 9.42·18-s − 1.83·19-s − 8.94·20-s − 5.23·21-s − 3.41·22-s − 1.45·23-s + 16.3·24-s − 25-s − 3.92·26-s + 3.84·27-s − 11.3·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 11^{4} \cdot 61^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 61 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 4 T + 21 T^{2} + 53 T^{3} + 159 T^{4} + 53 p T^{5} + 21 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 6 T + 5 p T^{2} + 121 T^{3} + 384 T^{4} + 121 p T^{5} + 5 p^{3} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 5 T + 40 T^{2} + 12 p T^{3} + 680 T^{4} + 12 p^{2} T^{5} + 40 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 10 T + 72 T^{2} + 332 T^{3} + 1521 T^{4} + 332 p T^{5} + 72 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 8 T + 77 T^{2} + 373 T^{3} + 2080 T^{4} + 373 p T^{5} + 77 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 7 T + 71 T^{2} + 397 T^{3} + 2311 T^{4} + 397 p T^{5} + 71 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 101 T^{2} + 309 T^{3} + 4178 T^{4} + 309 p T^{5} + 101 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 9 T + 74 T^{2} + 220 T^{3} + 1320 T^{4} + 220 p T^{5} + 74 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 11 T + 119 T^{2} + 827 T^{3} + 5511 T^{4} + 827 p T^{5} + 119 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 17 T + 229 T^{2} + 2170 T^{3} + 15438 T^{4} + 2170 p T^{5} + 229 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 11 T + 107 T^{2} + 921 T^{3} + 7617 T^{4} + 921 p T^{5} + 107 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 7 T + 156 T^{2} + 654 T^{3} + 9620 T^{4} + 654 p T^{5} + 156 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 16 T + 227 T^{2} - 2271 T^{3} + 17942 T^{4} - 2271 p T^{5} + 227 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 3 T + 116 T^{2} - 704 T^{3} + 8072 T^{4} - 704 p T^{5} + 116 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 5 T + 236 T^{2} - 796 T^{3} + 22402 T^{4} - 796 p T^{5} + 236 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 10 T + 199 T^{2} + 1427 T^{3} + 19545 T^{4} + 1427 p T^{5} + 199 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 9 T + 105 T^{2} + 741 T^{3} + 11403 T^{4} + 741 p T^{5} + 105 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 11 T + 278 T^{2} + 2478 T^{3} + 31770 T^{4} + 2478 p T^{5} + 278 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5 T + 269 T^{2} - 1083 T^{3} + 32009 T^{4} - 1083 p T^{5} + 269 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 8 T + 309 T^{2} - 1663 T^{3} + 38487 T^{4} - 1663 p T^{5} + 309 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 12 T + 289 T^{2} + 2645 T^{3} + 38579 T^{4} + 2645 p T^{5} + 289 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.51843832369090079520249354499, −5.99568862329156442200308120928, −5.90557040180825552084621218104, −5.74513808648770347739295810977, −5.69467984474894698434513285171, −5.07214645097961746030464434782, −4.96613608641697145730008799958, −4.92976604872709052147544154572, −4.78445908752774958921417645365, −4.31013703724824540958673281826, −4.23780504790073482663480606576, −4.08163457892087109561411239011, −3.86630835987166975103028435457, −3.67862220389413578861427490759, −3.54952864694099596095398532760, −3.40652406369662977764504013180, −3.39921135594231060865697817838, −2.76035917629323814000127050971, −2.75097080050230370483001723298, −2.66971518362161352447103546641, −2.34506872001969458374719879997, −2.07374107656679427706389619976, −1.74879753788453997010423519259, −1.71436565782694536087908154590, −1.67657419920329934636464807088, 0, 0, 0, 0,
1.67657419920329934636464807088, 1.71436565782694536087908154590, 1.74879753788453997010423519259, 2.07374107656679427706389619976, 2.34506872001969458374719879997, 2.66971518362161352447103546641, 2.75097080050230370483001723298, 2.76035917629323814000127050971, 3.39921135594231060865697817838, 3.40652406369662977764504013180, 3.54952864694099596095398532760, 3.67862220389413578861427490759, 3.86630835987166975103028435457, 4.08163457892087109561411239011, 4.23780504790073482663480606576, 4.31013703724824540958673281826, 4.78445908752774958921417645365, 4.92976604872709052147544154572, 4.96613608641697145730008799958, 5.07214645097961746030464434782, 5.69467984474894698434513285171, 5.74513808648770347739295810977, 5.90557040180825552084621218104, 5.99568862329156442200308120928, 6.51843832369090079520249354499
