| L(s) = 1 | − 2-s + 4-s − 8-s − 5·13-s + 16-s + 6·17-s + 4·19-s − 6·23-s − 5·25-s + 5·26-s + 6·29-s − 31-s − 32-s − 6·34-s − 37-s − 4·38-s − 6·41-s + 43-s + 6·46-s + 6·47-s + 5·50-s − 5·52-s + 6·53-s − 6·58-s + 6·59-s + 61-s + 62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.38·13-s + 1/4·16-s + 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s + 0.980·26-s + 1.11·29-s − 0.179·31-s − 0.176·32-s − 1.02·34-s − 0.164·37-s − 0.648·38-s − 0.937·41-s + 0.152·43-s + 0.884·46-s + 0.875·47-s + 0.707·50-s − 0.693·52-s + 0.824·53-s − 0.787·58-s + 0.781·59-s + 0.128·61-s + 0.127·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320166 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−12.65976385523824, −12.04890512559751, −12.01343161707542, −11.71267525530872, −10.96983804582755, −10.23805644247342, −10.07770866956384, −9.861055823240799, −9.267405694136267, −8.707108434350433, −8.239792969868909, −7.641066933093125, −7.489925462453465, −7.044976504822508, −6.273287618236131, −5.911063147114195, −5.302027945128930, −4.990085754213313, −4.204238230321306, −3.665258169796007, −3.114391860229807, −2.519075031174827, −2.039417482167811, −1.349642355211503, −0.7145542537960777, 0,
0.7145542537960777, 1.349642355211503, 2.039417482167811, 2.519075031174827, 3.114391860229807, 3.665258169796007, 4.204238230321306, 4.990085754213313, 5.302027945128930, 5.911063147114195, 6.273287618236131, 7.044976504822508, 7.489925462453465, 7.641066933093125, 8.239792969868909, 8.707108434350433, 9.267405694136267, 9.861055823240799, 10.07770866956384, 10.23805644247342, 10.96983804582755, 11.71267525530872, 12.01343161707542, 12.04890512559751, 12.65976385523824
