| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s − 3·11-s − 13-s + 15-s − 6·17-s + 2·19-s − 21-s + 3·23-s + 25-s − 5·27-s + 6·29-s + 7·31-s − 3·33-s − 35-s − 5·37-s − 39-s + 9·41-s + 2·43-s − 2·45-s + 9·47-s + 49-s − 6·51-s − 3·55-s + 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s − 0.904·11-s − 0.277·13-s + 0.258·15-s − 1.45·17-s + 0.458·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s − 0.962·27-s + 1.11·29-s + 1.25·31-s − 0.522·33-s − 0.169·35-s − 0.821·37-s − 0.160·39-s + 1.40·41-s + 0.304·43-s − 0.298·45-s + 1.31·47-s + 1/7·49-s − 0.840·51-s − 0.404·55-s + 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−15.49010395884248, −14.79748409039874, −14.42011789724977, −13.73108052585294, −13.32873416968377, −13.11203038976392, −12.16726411219123, −11.83952723242012, −11.03413500599177, −10.52881228527108, −10.11303581930614, −9.318009405164474, −8.912098073622320, −8.507573017527818, −7.760819138121431, −7.237803798608871, −6.540103860875090, −5.971655600792869, −5.353255407564357, −4.686271930450598, −4.067263772424243, −3.031902083969156, −2.683220178383304, −2.198118297246852, −1.013413972586316, 0,
1.013413972586316, 2.198118297246852, 2.683220178383304, 3.031902083969156, 4.067263772424243, 4.686271930450598, 5.353255407564357, 5.971655600792869, 6.540103860875090, 7.237803798608871, 7.760819138121431, 8.507573017527818, 8.912098073622320, 9.318009405164474, 10.11303581930614, 10.52881228527108, 11.03413500599177, 11.83952723242012, 12.16726411219123, 13.11203038976392, 13.32873416968377, 13.73108052585294, 14.42011789724977, 14.79748409039874, 15.49010395884248
