| L(s) = 1 | + 2·2-s − 2·3-s + 2·4-s − 4·5-s − 4·6-s + 4·7-s + 4·8-s − 8·10-s + 4·11-s − 4·12-s − 2·13-s + 8·14-s + 8·15-s + 3·16-s + 4·17-s − 8·20-s − 8·21-s + 8·22-s − 8·23-s − 8·24-s + 10·25-s − 4·26-s + 6·27-s + 8·28-s − 4·29-s + 16·30-s − 4·31-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s − 1.63·6-s + 1.51·7-s + 1.41·8-s − 2.52·10-s + 1.20·11-s − 1.15·12-s − 0.554·13-s + 2.13·14-s + 2.06·15-s + 3/4·16-s + 0.970·17-s − 1.78·20-s − 1.74·21-s + 1.70·22-s − 1.66·23-s − 1.63·24-s + 2·25-s − 0.784·26-s + 1.15·27-s + 1.51·28-s − 0.742·29-s + 2.92·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.421061030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.421061030\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + 9 T^{4} - p^{3} T^{5} + p^{3} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 2 T + 4 T^{2} + 2 T^{3} + 2 T^{4} + 2 p T^{5} + 4 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $S_4\times C_2$ | \( 1 - 4 T + 12 T^{2} - 36 T^{3} + 102 T^{4} - 36 p T^{5} + 12 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $\textrm{GL(2,3)}$ | \( 1 - 4 T + 28 T^{2} - 100 T^{3} + 422 T^{4} - 100 p T^{5} + 28 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 28 T^{2} + 46 T^{3} + 410 T^{4} + 46 p T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 4 T + 36 T^{2} - 188 T^{3} + 694 T^{4} - 188 p T^{5} + 36 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 8 T + 68 T^{2} + 376 T^{3} + 2358 T^{4} + 376 p T^{5} + 68 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $\textrm{GL(2,3)}$ | \( 1 + 4 T + 84 T^{2} + 332 T^{3} + 3238 T^{4} + 332 p T^{5} + 84 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 4 T + 44 T^{2} - 140 T^{3} + 166 T^{4} - 140 p T^{5} + 44 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 6 T + 124 T^{2} - 626 T^{3} + 6442 T^{4} - 626 p T^{5} + 124 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 220 T^{2} + 1936 T^{3} + 14438 T^{4} + 1936 p T^{5} + 220 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 4 T + 156 T^{2} - 468 T^{3} + 9750 T^{4} - 468 p T^{5} + 156 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 124 T^{2} + 1036 T^{3} + 8294 T^{4} + 1036 p T^{5} + 124 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 10 T + 4 p T^{2} - 1406 T^{3} + 16506 T^{4} - 1406 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 172 T^{2} + 224 T^{3} + 13142 T^{4} + 224 p T^{5} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 20 T + 300 T^{2} - 2972 T^{3} + 26502 T^{4} - 2972 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 18 T + 276 T^{2} - 3130 T^{3} + 26930 T^{4} - 3130 p T^{5} + 276 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 20 T + 316 T^{2} - 3236 T^{3} + 30566 T^{4} - 3236 p T^{5} + 316 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 28 T + 548 T^{2} - 6916 T^{3} + 69526 T^{4} - 6916 p T^{5} + 548 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 16 T + 348 T^{2} - 3312 T^{3} + 40646 T^{4} - 3312 p T^{5} + 348 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 260 T^{2} - 112 T^{3} + 29862 T^{4} - 112 p T^{5} + 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 4 T + 212 T^{2} + 1244 T^{3} + 22134 T^{4} + 1244 p T^{5} + 212 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 30 T + 612 T^{2} + 8738 T^{3} + 98522 T^{4} + 8738 p T^{5} + 612 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.61054074024909909076916988385, −6.55467413009441385029304705225, −5.99335280573231450040210584931, −5.86350402135829098367204157809, −5.65551111786547195104466626618, −5.39115809748685932343941940298, −5.09483183582300447916141946734, −5.07002820458782252301618911480, −5.00474620690255082167397730928, −4.73561164895114785653516741704, −4.45874657319188368051399772900, −4.11937921977363612826867595336, −3.93777180878089654692540093788, −3.89502231656176158638969992703, −3.65530108400113259204446361768, −3.50686542365179419950908498446, −3.21593653361583710890314896059, −2.57734648814662431764233080693, −2.57607082068416537821561297740, −1.97221363531484072848400416088, −1.79377811474673260537343947752, −1.78127667405207560327355660647, −1.04579847525429894475470299308, −0.71535563597378505722975222113, −0.42114664627238455157593270570,
0.42114664627238455157593270570, 0.71535563597378505722975222113, 1.04579847525429894475470299308, 1.78127667405207560327355660647, 1.79377811474673260537343947752, 1.97221363531484072848400416088, 2.57607082068416537821561297740, 2.57734648814662431764233080693, 3.21593653361583710890314896059, 3.50686542365179419950908498446, 3.65530108400113259204446361768, 3.89502231656176158638969992703, 3.93777180878089654692540093788, 4.11937921977363612826867595336, 4.45874657319188368051399772900, 4.73561164895114785653516741704, 5.00474620690255082167397730928, 5.07002820458782252301618911480, 5.09483183582300447916141946734, 5.39115809748685932343941940298, 5.65551111786547195104466626618, 5.86350402135829098367204157809, 5.99335280573231450040210584931, 6.55467413009441385029304705225, 6.61054074024909909076916988385
