| L(s) = 1 | + 5-s − 2·7-s − 6·11-s − 2·17-s + 4·23-s + 25-s − 8·29-s − 8·31-s − 2·35-s + 4·37-s − 4·41-s + 6·43-s − 12·47-s − 3·49-s + 6·53-s − 6·55-s + 4·59-s + 2·61-s − 8·67-s + 6·73-s + 12·77-s + 8·79-s + 4·83-s − 2·85-s − 4·89-s − 12·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 1.80·11-s − 0.485·17-s + 0.834·23-s + 1/5·25-s − 1.48·29-s − 1.43·31-s − 0.338·35-s + 0.657·37-s − 0.624·41-s + 0.914·43-s − 1.75·47-s − 3/7·49-s + 0.824·53-s − 0.809·55-s + 0.520·59-s + 0.256·61-s − 0.977·67-s + 0.702·73-s + 1.36·77-s + 0.900·79-s + 0.439·83-s − 0.216·85-s − 0.423·89-s − 1.21·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.06491097956793, −12.81713607802116, −12.34010897398294, −11.49210498810547, −11.11325462462628, −10.81034521407415, −10.22072113089780, −9.835094250185401, −9.383380586838324, −8.954152863386048, −8.391410687150889, −7.880292836317569, −7.263503905490880, −7.112175421224633, −6.323873639921828, −5.889692866972130, −5.411588580277621, −4.982994149328783, −4.494178854590939, −3.528523601078673, −3.383671581576343, −2.580973820695841, −2.215610596653915, −1.593164962990099, −0.5891656346847220, 0,
0.5891656346847220, 1.593164962990099, 2.215610596653915, 2.580973820695841, 3.383671581576343, 3.528523601078673, 4.494178854590939, 4.982994149328783, 5.411588580277621, 5.889692866972130, 6.323873639921828, 7.112175421224633, 7.263503905490880, 7.880292836317569, 8.391410687150889, 8.954152863386048, 9.383380586838324, 9.835094250185401, 10.22072113089780, 10.81034521407415, 11.11325462462628, 11.49210498810547, 12.34010897398294, 12.81713607802116, 13.06491097956793
