| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 15-s + 16-s − 18-s − 4·19-s + 20-s − 7·23-s + 24-s + 25-s − 27-s − 5·29-s + 30-s + 6·31-s − 32-s + 36-s − 4·37-s + 4·38-s − 40-s − 5·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.223·20-s − 1.45·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 0.182·30-s + 1.07·31-s − 0.176·32-s + 1/6·36-s − 0.657·37-s + 0.648·38-s − 0.158·40-s − 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−13.48782277648287, −13.20271492190992, −12.63253236200185, −12.02812763596468, −11.78689975476495, −11.23798085240788, −10.75604604194978, −10.21064834886497, −9.914226931207047, −9.579854082133490, −8.827247925572858, −8.319287848270339, −8.097660741167904, −7.346651364975692, −6.806352489167282, −6.374436813372869, −6.036028605406988, −5.378461829154240, −4.881555961699361, −4.278797847852475, −3.609459441653190, −3.034398401772201, −2.197494193696983, −1.760093159163311, −1.242007305643967, 0, 0,
1.242007305643967, 1.760093159163311, 2.197494193696983, 3.034398401772201, 3.609459441653190, 4.278797847852475, 4.881555961699361, 5.378461829154240, 6.036028605406988, 6.374436813372869, 6.806352489167282, 7.346651364975692, 8.097660741167904, 8.319287848270339, 8.827247925572858, 9.579854082133490, 9.914226931207047, 10.21064834886497, 10.75604604194978, 11.23798085240788, 11.78689975476495, 12.02812763596468, 12.63253236200185, 13.20271492190992, 13.48782277648287
