| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 4·5-s + 16·6-s − 7-s − 20·8-s + 10·9-s − 16·10-s − 4·11-s − 40·12-s + 3·13-s + 4·14-s − 16·15-s + 35·16-s + 4·17-s − 40·18-s − 9·19-s + 40·20-s + 4·21-s + 16·22-s − 23-s + 80·24-s + 10·25-s − 12·26-s − 20·27-s − 10·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 1.78·5-s + 6.53·6-s − 0.377·7-s − 7.07·8-s + 10/3·9-s − 5.05·10-s − 1.20·11-s − 11.5·12-s + 0.832·13-s + 1.06·14-s − 4.13·15-s + 35/4·16-s + 0.970·17-s − 9.42·18-s − 2.06·19-s + 8.94·20-s + 0.872·21-s + 3.41·22-s − 0.208·23-s + 16.3·24-s + 2·25-s − 2.35·26-s − 3.84·27-s − 1.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 11^{4} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + T + 20 T^{2} + 17 T^{3} + 190 T^{4} + 17 p T^{5} + 20 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 3 T + 10 T^{2} + 11 T^{3} - 166 T^{4} + 11 p T^{5} + 10 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 9 T + 98 T^{2} + 521 T^{3} + 2994 T^{4} + 521 p T^{5} + 98 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + T + 2 p T^{2} + 53 T^{3} + 1546 T^{4} + 53 p T^{5} + 2 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 8 T + 60 T^{2} + 344 T^{3} + 2390 T^{4} + 344 p T^{5} + 60 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + T - 40 T^{2} + 29 T^{3} + 2062 T^{4} + 29 p T^{5} - 40 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - T + 46 T^{2} - 411 T^{3} + 26 p T^{4} - 411 p T^{5} + 46 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 2 T + 132 T^{2} - 214 T^{3} + 7590 T^{4} - 214 p T^{5} + 132 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 4 T + 80 T^{2} - 92 T^{3} + 3694 T^{4} - 92 p T^{5} + 80 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 2 T + 108 T^{2} + 6 p T^{3} + 6758 T^{4} + 6 p^{2} T^{5} + 108 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T + 160 T^{2} - 858 T^{3} + 11326 T^{4} - 858 p T^{5} + 160 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 24 T + 384 T^{2} + 4176 T^{3} + 36846 T^{4} + 4176 p T^{5} + 384 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 7 T + 242 T^{2} + 1221 T^{3} + 22074 T^{4} + 1221 p T^{5} + 242 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 11 T + 242 T^{2} - 2135 T^{3} + 23538 T^{4} - 2135 p T^{5} + 242 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 22 T + 416 T^{2} + 4638 T^{3} + 47342 T^{4} + 4638 p T^{5} + 416 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 14 T + 212 T^{2} - 1522 T^{3} + 16006 T^{4} - 1522 p T^{5} + 212 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 12 T + 272 T^{2} + 2132 T^{3} + 29838 T^{4} + 2132 p T^{5} + 272 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 25 T + 484 T^{2} + 6353 T^{3} + 66054 T^{4} + 6353 p T^{5} + 484 p^{2} T^{6} + 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 20 T + 4 p T^{2} + 4620 T^{3} + 48278 T^{4} + 4620 p T^{5} + 4 p^{3} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 19 T + 404 T^{2} - 4393 T^{3} + 55782 T^{4} - 4393 p T^{5} + 404 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.17665684527435945504969855105, −6.05977349946745963283654729286, −5.95907873237781559422093892237, −5.67960620921647646851454692187, −5.63567601972072327880230069485, −5.06839171986064613232782194933, −5.05071898525166716244604594132, −5.04334023657694671118031898863, −5.02191071438888043995957008863, −4.28688636982346458976974836894, −4.18097866820459678706433041273, −4.02113793673208470362785350168, −3.96315752068079998601762069440, −3.24641724600155060796389946618, −3.11628490451257282004335490414, −3.10441271218852292618293175856, −2.90721942467223289599764641904, −2.25924816925393442186214892914, −2.24728474008360345050848771900, −2.06182172534747779152327571110, −1.95667895141289558191209022630, −1.37838481067971724126154070517, −1.23625692498825687582583379421, −1.18684149150279063214157711910, −1.10857706494769741032145265095, 0, 0, 0, 0,
1.10857706494769741032145265095, 1.18684149150279063214157711910, 1.23625692498825687582583379421, 1.37838481067971724126154070517, 1.95667895141289558191209022630, 2.06182172534747779152327571110, 2.24728474008360345050848771900, 2.25924816925393442186214892914, 2.90721942467223289599764641904, 3.10441271218852292618293175856, 3.11628490451257282004335490414, 3.24641724600155060796389946618, 3.96315752068079998601762069440, 4.02113793673208470362785350168, 4.18097866820459678706433041273, 4.28688636982346458976974836894, 5.02191071438888043995957008863, 5.04334023657694671118031898863, 5.05071898525166716244604594132, 5.06839171986064613232782194933, 5.63567601972072327880230069485, 5.67960620921647646851454692187, 5.95907873237781559422093892237, 6.05977349946745963283654729286, 6.17665684527435945504969855105
