| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s − 3·9-s − 2·13-s − 4·14-s − 16-s + 17-s + 3·18-s + 4·19-s − 4·23-s + 2·26-s − 4·28-s − 6·29-s + 4·31-s − 5·32-s − 34-s + 3·36-s + 2·37-s − 4·38-s + 6·41-s + 4·43-s + 4·46-s + 9·49-s + 2·52-s − 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s − 9-s − 0.554·13-s − 1.06·14-s − 1/4·16-s + 0.242·17-s + 0.707·18-s + 0.917·19-s − 0.834·23-s + 0.392·26-s − 0.755·28-s − 1.11·29-s + 0.718·31-s − 0.883·32-s − 0.171·34-s + 1/2·36-s + 0.328·37-s − 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.589·46-s + 9/7·49-s + 0.277·52-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−14.51208444001560, −14.29675986291402, −13.94149304666488, −13.31862439985622, −12.68083711023853, −11.99650984927075, −11.56214108240005, −11.16131101141660, −10.59751145478778, −10.05507257160231, −9.377361559999138, −9.116430766958097, −8.300821220728159, −8.117054859056938, −7.521348616471706, −7.208520869472250, −5.949678720652368, −5.715074210482257, −4.919973397505425, −4.605266775829156, −3.908605924016859, −3.076703723600732, −2.279241292303657, −1.607976447521990, −0.8995093476066003, 0,
0.8995093476066003, 1.607976447521990, 2.279241292303657, 3.076703723600732, 3.908605924016859, 4.605266775829156, 4.919973397505425, 5.715074210482257, 5.949678720652368, 7.208520869472250, 7.521348616471706, 8.117054859056938, 8.300821220728159, 9.116430766958097, 9.377361559999138, 10.05507257160231, 10.59751145478778, 11.16131101141660, 11.56214108240005, 11.99650984927075, 12.68083711023853, 13.31862439985622, 13.94149304666488, 14.29675986291402, 14.51208444001560
