| L(s) = 1 | − 2-s + 4-s − 3·5-s − 7-s − 8-s + 3·10-s − 5·13-s + 14-s + 16-s − 3·17-s + 4·19-s − 3·20-s − 3·23-s + 4·25-s + 5·26-s − 28-s + 8·31-s − 32-s + 3·34-s + 3·35-s − 4·37-s − 4·38-s + 3·40-s + 3·41-s − 2·43-s + 3·46-s + 49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.377·7-s − 0.353·8-s + 0.948·10-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s + 0.917·19-s − 0.670·20-s − 0.625·23-s + 4/5·25-s + 0.980·26-s − 0.188·28-s + 1.43·31-s − 0.176·32-s + 0.514·34-s + 0.507·35-s − 0.657·37-s − 0.648·38-s + 0.474·40-s + 0.468·41-s − 0.304·43-s + 0.442·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.12101085222322, −14.43778386814859, −13.99348563135150, −13.24591858184135, −12.64739056176963, −12.09379693188478, −11.73205227714996, −11.44138442050323, −10.64887560283025, −10.10609994486402, −9.725545005632062, −9.083241610141034, −8.502472737637960, −7.948623055775432, −7.571181676961625, −6.956542659770563, −6.640133808920094, −5.718096031910293, −5.079864801052484, −4.337202550838692, −3.921151382793117, −2.976842997676949, −2.656692544587109, −1.678297913192811, −0.6511300916324188, 0,
0.6511300916324188, 1.678297913192811, 2.656692544587109, 2.976842997676949, 3.921151382793117, 4.337202550838692, 5.079864801052484, 5.718096031910293, 6.640133808920094, 6.956542659770563, 7.571181676961625, 7.948623055775432, 8.502472737637960, 9.083241610141034, 9.725545005632062, 10.10609994486402, 10.64887560283025, 11.44138442050323, 11.73205227714996, 12.09379693188478, 12.64739056176963, 13.24591858184135, 13.99348563135150, 14.43778386814859, 15.12101085222322
