| L(s) = 1 | − 2-s + 2·3-s + 4-s + 5-s − 2·6-s − 4·7-s − 8-s + 9-s − 10-s + 2·12-s − 13-s + 4·14-s + 2·15-s + 16-s + 2·17-s − 18-s + 20-s − 8·21-s − 4·23-s − 2·24-s + 25-s + 26-s − 4·27-s − 4·28-s + 8·29-s − 2·30-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.447·5-s − 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.577·12-s − 0.277·13-s + 1.06·14-s + 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 1.74·21-s − 0.834·23-s − 0.408·24-s + 1/5·25-s + 0.196·26-s − 0.769·27-s − 0.755·28-s + 1.48·29-s − 0.365·30-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−16.17023486654123, −15.92181563594788, −15.10875756550504, −14.65786266346165, −14.06119582878371, −13.50994421749752, −13.07186875156220, −12.30919873799214, −11.98881824827053, −10.97346421697515, −10.36411849576777, −9.723673929606147, −9.503835227479962, −9.025339386488169, −8.138374964863867, −7.922664221324024, −7.062105899563075, −6.347986675261295, −6.056254801655416, −5.050838961608231, −4.005038718165469, −3.317779338690648, −2.741955730883494, −2.251605864937002, −1.144881349036892, 0,
1.144881349036892, 2.251605864937002, 2.741955730883494, 3.317779338690648, 4.005038718165469, 5.050838961608231, 6.056254801655416, 6.347986675261295, 7.062105899563075, 7.922664221324024, 8.138374964863867, 9.025339386488169, 9.503835227479962, 9.723673929606147, 10.36411849576777, 10.97346421697515, 11.98881824827053, 12.30919873799214, 13.07186875156220, 13.50994421749752, 14.06119582878371, 14.65786266346165, 15.10875756550504, 15.92181563594788, 16.17023486654123
