| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 7-s − 8-s + 9-s − 2·10-s + 2·11-s − 2·12-s − 4·13-s + 14-s − 4·15-s + 16-s − 6·17-s − 18-s + 6·19-s + 2·20-s + 2·21-s − 2·22-s + 8·23-s + 2·24-s − 25-s + 4·26-s + 4·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.603·11-s − 0.577·12-s − 1.10·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.426·22-s + 1.66·23-s + 0.408·24-s − 1/5·25-s + 0.784·26-s + 0.769·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 41 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−15.92621024277975, −15.27510125197473, −14.72760211947996, −14.15789547409879, −13.39099487845982, −12.99246148152247, −12.34008295637357, −11.72493579576174, −11.32950267439542, −10.85820037194294, −10.19231971049679, −9.647342019705777, −9.230400244587524, −8.803679366093771, −7.799437227697868, −7.087606584910038, −6.736432319874429, −6.159968268541605, −5.545395336746705, −5.009404565007561, −4.380648512158708, −3.176913660508028, −2.599386714246184, −1.709011078576505, −0.8976425583126328, 0,
0.8976425583126328, 1.709011078576505, 2.599386714246184, 3.176913660508028, 4.380648512158708, 5.009404565007561, 5.545395336746705, 6.159968268541605, 6.736432319874429, 7.087606584910038, 7.799437227697868, 8.803679366093771, 9.230400244587524, 9.647342019705777, 10.19231971049679, 10.85820037194294, 11.32950267439542, 11.72493579576174, 12.34008295637357, 12.99246148152247, 13.39099487845982, 14.15789547409879, 14.72760211947996, 15.27510125197473, 15.92621024277975
