| L(s) = 1 | − 2·4-s + 5-s − 4·7-s + 3·11-s + 13-s + 4·16-s − 19-s − 2·20-s − 6·23-s + 25-s + 8·28-s − 3·29-s + 8·31-s − 4·35-s + 2·37-s − 9·41-s + 8·43-s − 6·44-s − 12·47-s + 9·49-s − 2·52-s + 12·53-s + 3·55-s − 3·59-s + 5·61-s − 8·64-s + 65-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s − 1.51·7-s + 0.904·11-s + 0.277·13-s + 16-s − 0.229·19-s − 0.447·20-s − 1.25·23-s + 1/5·25-s + 1.51·28-s − 0.557·29-s + 1.43·31-s − 0.676·35-s + 0.328·37-s − 1.40·41-s + 1.21·43-s − 0.904·44-s − 1.75·47-s + 9/7·49-s − 0.277·52-s + 1.64·53-s + 0.404·55-s − 0.390·59-s + 0.640·61-s − 64-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.50175599250418, −13.10045368116932, −12.53974396998950, −12.21504106389305, −11.68871378687923, −11.10760543977165, −10.24894685360018, −10.07276985163934, −9.676513302400942, −9.265201208512454, −8.730393661538865, −8.341712335858693, −7.774591027015692, −6.986567304736900, −6.534931843185478, −6.161110019951288, −5.685736296321041, −5.120216839997483, −4.337381250730303, −3.985465009233711, −3.480198022954554, −2.941460521083557, −2.204890297690559, −1.414142707537724, −0.7050389184604936, 0,
0.7050389184604936, 1.414142707537724, 2.204890297690559, 2.941460521083557, 3.480198022954554, 3.985465009233711, 4.337381250730303, 5.120216839997483, 5.685736296321041, 6.161110019951288, 6.534931843185478, 6.986567304736900, 7.774591027015692, 8.341712335858693, 8.730393661538865, 9.265201208512454, 9.676513302400942, 10.07276985163934, 10.24894685360018, 11.10760543977165, 11.68871378687923, 12.21504106389305, 12.53974396998950, 13.10045368116932, 13.50175599250418
