| L(s) = 1 | + 3-s − 2·5-s − 7-s + 9-s + 4·11-s + 13-s − 2·15-s − 6·17-s − 4·19-s − 21-s − 25-s + 27-s + 6·29-s + 4·33-s + 2·35-s + 6·37-s + 39-s − 6·41-s − 4·43-s − 2·45-s + 49-s − 6·51-s − 2·53-s − 8·55-s − 4·57-s − 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.516·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.696·33-s + 0.338·35-s + 0.986·37-s + 0.160·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s + 1/7·49-s − 0.840·51-s − 0.274·53-s − 1.07·55-s − 0.529·57-s − 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−8.218109609397808997746007392570, −7.22655508589695350080520557108, −6.63185189155182396691780576676, −6.07365909087589362413070795141, −4.61882441072644281753268775610, −4.19903354850439113622039402635, −3.47052781277382618078647839299, −2.53911424631192626116632529386, −1.44637528951517179318251308850, 0,
1.44637528951517179318251308850, 2.53911424631192626116632529386, 3.47052781277382618078647839299, 4.19903354850439113622039402635, 4.61882441072644281753268775610, 6.07365909087589362413070795141, 6.63185189155182396691780576676, 7.22655508589695350080520557108, 8.218109609397808997746007392570
