| L(s) = 1 | + 2·7-s + 4·11-s − 2·13-s − 2·19-s − 5·25-s + 6·29-s − 6·37-s + 6·41-s − 10·43-s − 3·49-s − 4·53-s − 4·59-s − 6·61-s + 2·67-s + 8·71-s + 2·73-s + 8·77-s − 2·79-s + 12·83-s − 4·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s − 0.554·13-s − 0.458·19-s − 25-s + 1.11·29-s − 0.986·37-s + 0.937·41-s − 1.52·43-s − 3/7·49-s − 0.549·53-s − 0.520·59-s − 0.768·61-s + 0.244·67-s + 0.949·71-s + 0.234·73-s + 0.911·77-s − 0.225·79-s + 1.31·83-s − 0.423·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.78376075944762, −12.28917934050585, −12.06449562034048, −11.48165617132129, −11.21929332418555, −10.65703833157432, −10.06615623557261, −9.774915093595969, −9.187493300100352, −8.738984772525482, −8.320803161035476, −7.796096592791929, −7.397683246912410, −6.757654739344642, −6.346027541404667, −5.985516046687651, −5.163505855321093, −4.814766019561464, −4.393713551477749, −3.726562298953374, −3.351004748665654, −2.563322547083826, −1.925879001046141, −1.556010565473276, −0.8297726632970706, 0,
0.8297726632970706, 1.556010565473276, 1.925879001046141, 2.563322547083826, 3.351004748665654, 3.726562298953374, 4.393713551477749, 4.814766019561464, 5.163505855321093, 5.985516046687651, 6.346027541404667, 6.757654739344642, 7.397683246912410, 7.796096592791929, 8.320803161035476, 8.738984772525482, 9.187493300100352, 9.774915093595969, 10.06615623557261, 10.65703833157432, 11.21929332418555, 11.48165617132129, 12.06449562034048, 12.28917934050585, 12.78376075944762
