| L(s) = 1 | + 2·5-s + 2·7-s + 4·11-s + 2·13-s + 2·17-s − 23-s − 25-s − 4·29-s + 4·35-s − 2·37-s − 2·43-s + 12·47-s − 3·49-s + 2·53-s + 8·55-s − 12·59-s − 14·61-s + 4·65-s − 2·67-s + 6·73-s + 8·77-s − 6·79-s + 4·83-s + 4·85-s + 18·89-s + 4·91-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.208·23-s − 1/5·25-s − 0.742·29-s + 0.676·35-s − 0.328·37-s − 0.304·43-s + 1.75·47-s − 3/7·49-s + 0.274·53-s + 1.07·55-s − 1.56·59-s − 1.79·61-s + 0.496·65-s − 0.244·67-s + 0.702·73-s + 0.911·77-s − 0.675·79-s + 0.439·83-s + 0.433·85-s + 1.90·89-s + 0.419·91-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 298908 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 298908 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 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.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−12.95095867024584, −12.34003292842190, −12.01516976571999, −11.60839717077058, −11.02399857980124, −10.66045992173144, −10.24047616884072, −9.545557992596505, −9.269085796629659, −8.925784044306098, −8.315596645698354, −7.787961108837235, −7.412926180087988, −6.722074262461877, −6.321046617085677, −5.691506090247254, −5.640863408790341, −4.706839082760582, −4.445685599897852, −3.652034927696968, −3.398111504510288, −2.494086555126374, −1.955470217390890, −1.422606207191134, −1.083267786067296, 0,
1.083267786067296, 1.422606207191134, 1.955470217390890, 2.494086555126374, 3.398111504510288, 3.652034927696968, 4.445685599897852, 4.706839082760582, 5.640863408790341, 5.691506090247254, 6.321046617085677, 6.722074262461877, 7.412926180087988, 7.787961108837235, 8.315596645698354, 8.925784044306098, 9.269085796629659, 9.545557992596505, 10.24047616884072, 10.66045992173144, 11.02399857980124, 11.60839717077058, 12.01516976571999, 12.34003292842190, 12.95095867024584
