| L(s) = 1 | − 5-s + 2·11-s − 8·17-s − 4·19-s + 25-s − 6·29-s + 2·31-s + 4·37-s + 6·41-s − 4·43-s + 6·53-s − 2·55-s − 2·59-s − 10·61-s + 4·67-s + 12·71-s + 14·73-s + 10·79-s + 4·83-s + 8·85-s + 6·89-s + 4·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 1.94·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.657·37-s + 0.937·41-s − 0.609·43-s + 0.824·53-s − 0.269·55-s − 0.260·59-s − 1.28·61-s + 0.488·67-s + 1.42·71-s + 1.63·73-s + 1.12·79-s + 0.439·83-s + 0.867·85-s + 0.635·89-s + 0.410·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 282240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 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.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 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 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.01089198263735, −12.39412158255526, −12.19619899078785, −11.42426958002465, −11.14166913234426, −10.85955659597179, −10.33984154952186, −9.521269433239859, −9.327521950872304, −8.836849823808498, −8.334234333045157, −7.910251658392812, −7.357329256300145, −6.755165614760991, −6.457964461736315, −6.059039073840007, −5.266093049150790, −4.783656471582889, −4.222010957326411, −3.917510958502405, −3.370479213318438, −2.406857447915067, −2.275826361897731, −1.475577618368282, −0.6642392167847912, 0,
0.6642392167847912, 1.475577618368282, 2.275826361897731, 2.406857447915067, 3.370479213318438, 3.917510958502405, 4.222010957326411, 4.783656471582889, 5.266093049150790, 6.059039073840007, 6.457964461736315, 6.755165614760991, 7.357329256300145, 7.910251658392812, 8.334234333045157, 8.836849823808498, 9.327521950872304, 9.521269433239859, 10.33984154952186, 10.85955659597179, 11.14166913234426, 11.42426958002465, 12.19619899078785, 12.39412158255526, 13.01089198263735
