| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s − 13-s − 4·14-s + 16-s + 6·17-s − 4·19-s − 6·23-s + 26-s + 4·28-s − 6·29-s − 10·31-s − 32-s − 6·34-s + 10·37-s + 4·38-s − 6·41-s + 4·43-s + 6·46-s − 12·47-s + 9·49-s − 52-s − 12·53-s − 4·56-s + 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s − 0.277·13-s − 1.06·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 0.196·26-s + 0.755·28-s − 1.11·29-s − 1.79·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.884·46-s − 1.75·47-s + 9/7·49-s − 0.138·52-s − 1.64·53-s − 0.534·56-s + 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−7.908623718672730424328179759501, −7.42255676240836548270454419061, −6.31753294995797634546418099914, −5.66825166690823221850353693800, −4.90456128216973321171866465162, −4.08934514629340824000125194626, −3.12953578086723917615061490149, −1.91306337254860114149678235658, −1.51795954928522912559953276172, 0,
1.51795954928522912559953276172, 1.91306337254860114149678235658, 3.12953578086723917615061490149, 4.08934514629340824000125194626, 4.90456128216973321171866465162, 5.66825166690823221850353693800, 6.31753294995797634546418099914, 7.42255676240836548270454419061, 7.908623718672730424328179759501
