| L(s) = 1 | − 2·2-s + 3·3-s + 44·5-s − 6·6-s − 7·7-s + 8·8-s − 88·10-s − 16·11-s + 91·13-s + 14·14-s + 132·15-s − 16·16-s + 99·17-s + 22·19-s − 21·21-s + 32·22-s − 153·23-s + 24·24-s + 1.20e3·25-s − 182·26-s − 27·27-s − 222·29-s − 264·30-s + 182·31-s − 48·33-s − 198·34-s − 308·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 3.93·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2.78·10-s − 0.438·11-s + 1.94·13-s + 0.267·14-s + 2.27·15-s − 1/4·16-s + 1.41·17-s + 0.265·19-s − 0.218·21-s + 0.310·22-s − 1.38·23-s + 0.204·24-s + 9.61·25-s − 1.37·26-s − 0.192·27-s − 1.42·29-s − 1.60·30-s + 1.05·31-s − 0.253·33-s − 0.998·34-s − 1.48·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298116 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.152051074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.152051074\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 7 p T + p^{3} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 22 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 16 T - 1075 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 99 T + 4888 T^{2} - 99 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T - 6375 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 153 T + 11242 T^{2} + 153 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 222 T + 24895 T^{2} + 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 266 T + 20103 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 378 T + 73963 T^{2} - 378 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 85 T - 72282 T^{2} + 85 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 262 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 7 p T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 515 T + 59846 T^{2} + 515 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 483 T + 6308 T^{2} + 483 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 155 T - 276738 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 849 T + 362890 T^{2} - 849 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 284 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 116 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 323 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 537 T - 416600 T^{2} + 537 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 892 T - 117009 T^{2} + 892 p^{3} T^{3} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.19346896092073684303120209908, −10.07829863158123358199235413281, −9.632061930947750656436341572558, −9.514794734678228818953473809976, −8.905891401573734055014846366794, −8.783972077000982732390476669093, −7.999868453354922183931629732675, −7.74827720145902738869162626763, −6.72959360174637360251911509134, −6.40293416707224975160559555569, −5.96605270237494583686473589707, −5.71359397433451210606528334101, −5.38733462904612682929608695598, −4.65219524793157897230766611049, −3.64450512169865314240829137527, −3.04598135996937827272461945861, −2.38026550489331008153648201260, −2.01123501734342055054080408903, −1.28217631351688089116720078464, −1.01713968434453822484568122939,
1.01713968434453822484568122939, 1.28217631351688089116720078464, 2.01123501734342055054080408903, 2.38026550489331008153648201260, 3.04598135996937827272461945861, 3.64450512169865314240829137527, 4.65219524793157897230766611049, 5.38733462904612682929608695598, 5.71359397433451210606528334101, 5.96605270237494583686473589707, 6.40293416707224975160559555569, 6.72959360174637360251911509134, 7.74827720145902738869162626763, 7.999868453354922183931629732675, 8.783972077000982732390476669093, 8.905891401573734055014846366794, 9.514794734678228818953473809976, 9.632061930947750656436341572558, 10.07829863158123358199235413281, 10.19346896092073684303120209908
