| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 2·13-s + 14-s + 16-s − 6·17-s + 4·19-s + 2·26-s + 28-s + 6·29-s − 4·31-s + 32-s − 6·34-s − 2·37-s + 4·38-s + 6·41-s − 4·43-s + 49-s + 2·52-s + 6·53-s + 56-s + 6·58-s − 2·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 0.554·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.392·26-s + 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.328·37-s + 0.648·38-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.277·52-s + 0.824·53-s + 0.133·56-s + 0.787·58-s − 0.256·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 381150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−12.84122589030732, −12.28669170709322, −11.72207512741037, −11.39423965229902, −11.08068985509072, −10.54730881700359, −10.10014181885876, −9.594947316619490, −8.897191422750364, −8.689929998903928, −8.153774844942225, −7.522157757516389, −7.149410364631832, −6.672214923537275, −6.169456105360833, −5.727455871673025, −5.147089596163883, −4.763720764476649, −4.204670683301551, −3.812308782243384, −3.177903702890233, −2.636059409835047, −2.131724180085461, −1.470863649652887, −0.9190230535064484, 0,
0.9190230535064484, 1.470863649652887, 2.131724180085461, 2.636059409835047, 3.177903702890233, 3.812308782243384, 4.204670683301551, 4.763720764476649, 5.147089596163883, 5.727455871673025, 6.169456105360833, 6.672214923537275, 7.149410364631832, 7.522157757516389, 8.153774844942225, 8.689929998903928, 8.897191422750364, 9.594947316619490, 10.10014181885876, 10.54730881700359, 11.08068985509072, 11.39423965229902, 11.72207512741037, 12.28669170709322, 12.84122589030732
