| L(s) = 1 | + 3·5-s + 2·11-s − 5·13-s + 4·17-s − 6·19-s + 23-s + 4·25-s − 29-s + 4·31-s + 37-s − 41-s − 43-s − 9·47-s + 6·53-s + 6·55-s + 8·59-s − 12·61-s − 15·65-s + 16·67-s − 10·73-s − 6·79-s + 12·83-s + 12·85-s + 4·89-s − 18·95-s − 3·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.603·11-s − 1.38·13-s + 0.970·17-s − 1.37·19-s + 0.208·23-s + 4/5·25-s − 0.185·29-s + 0.718·31-s + 0.164·37-s − 0.156·41-s − 0.152·43-s − 1.31·47-s + 0.824·53-s + 0.809·55-s + 1.04·59-s − 1.53·61-s − 1.86·65-s + 1.95·67-s − 1.17·73-s − 0.675·79-s + 1.31·83-s + 1.30·85-s + 0.423·89-s − 1.84·95-s − 0.304·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−13.39628767626661, −13.14619985328668, −12.54688031311905, −12.13149542293981, −11.73560185362900, −11.06965172032066, −10.46721328656677, −10.09412535732214, −9.758756692492660, −9.266383979548802, −8.874412189900829, −8.101533738951403, −7.837525073938409, −6.956111030555464, −6.654170909003294, −6.234765477220872, −5.420550873564372, −5.355668044640532, −4.535075921093319, −4.113528247800099, −3.271876799360851, −2.694149995069236, −2.146436064720302, −1.666577303170094, −0.9408446274528168, 0,
0.9408446274528168, 1.666577303170094, 2.146436064720302, 2.694149995069236, 3.271876799360851, 4.113528247800099, 4.535075921093319, 5.355668044640532, 5.420550873564372, 6.234765477220872, 6.654170909003294, 6.956111030555464, 7.837525073938409, 8.101533738951403, 8.874412189900829, 9.266383979548802, 9.758756692492660, 10.09412535732214, 10.46721328656677, 11.06965172032066, 11.73560185362900, 12.13149542293981, 12.54688031311905, 13.14619985328668, 13.39628767626661
