| L(s) = 1 | − 2-s − 4-s − 2·5-s − 7-s + 3·8-s + 2·10-s + 11-s + 13-s + 14-s − 16-s − 17-s + 2·20-s − 22-s − 4·23-s − 25-s − 26-s + 28-s − 5·29-s − 5·31-s − 5·32-s + 34-s + 2·35-s − 6·37-s − 6·40-s + 2·41-s + 4·43-s − 44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s + 1.06·8-s + 0.632·10-s + 0.301·11-s + 0.277·13-s + 0.267·14-s − 1/4·16-s − 0.242·17-s + 0.447·20-s − 0.213·22-s − 0.834·23-s − 1/5·25-s − 0.196·26-s + 0.188·28-s − 0.928·29-s − 0.898·31-s − 0.883·32-s + 0.171·34-s + 0.338·35-s − 0.986·37-s − 0.948·40-s + 0.312·41-s + 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 295659 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.21410554105170, −12.55725043338720, −12.24135498733141, −11.84380458990774, −11.14426091691418, −10.78529789979111, −10.53540695352063, −9.727526624710252, −9.477101015948464, −9.009363682894327, −8.624468384382896, −8.014875652572573, −7.714845514059351, −7.291518424686361, −6.758751922992473, −6.160167982820901, −5.536315486496599, −5.166858802809229, −4.318941676849298, −4.005255574863378, −3.718927783440086, −3.023002061738169, −2.197942012227590, −1.620168787404002, −0.9678527013388691, 0, 0,
0.9678527013388691, 1.620168787404002, 2.197942012227590, 3.023002061738169, 3.718927783440086, 4.005255574863378, 4.318941676849298, 5.166858802809229, 5.536315486496599, 6.160167982820901, 6.758751922992473, 7.291518424686361, 7.714845514059351, 8.014875652572573, 8.624468384382896, 9.009363682894327, 9.477101015948464, 9.727526624710252, 10.53540695352063, 10.78529789979111, 11.14426091691418, 11.84380458990774, 12.24135498733141, 12.55725043338720, 13.21410554105170
