| L(s) = 1 | − 2·4-s + 2·7-s + 2·13-s + 4·16-s − 7·17-s + 6·19-s − 4·28-s − 7·29-s + 3·37-s − 41-s − 5·43-s − 3·49-s − 4·52-s − 7·53-s − 14·59-s + 61-s − 8·64-s + 2·67-s + 14·68-s + 7·71-s + 3·73-s − 12·76-s + 10·79-s + 14·83-s − 7·89-s + 4·91-s − 12·97-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s + 0.554·13-s + 16-s − 1.69·17-s + 1.37·19-s − 0.755·28-s − 1.29·29-s + 0.493·37-s − 0.156·41-s − 0.762·43-s − 3/7·49-s − 0.554·52-s − 0.961·53-s − 1.82·59-s + 0.128·61-s − 64-s + 0.244·67-s + 1.69·68-s + 0.830·71-s + 0.351·73-s − 1.37·76-s + 1.12·79-s + 1.53·83-s − 0.741·89-s + 0.419·91-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 7 T + p T^{2} \) | 1.89.h |
| 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
−7.61122226004038614019204616525, −6.65701861079484676778234933882, −5.93665291599261526039250972586, −5.08354147988145076291484449852, −4.72976238457494919343474733063, −3.89159705681680618594694693660, −3.23864990114924671493142556091, −2.04968671750905412037886780951, −1.19014502772381063096931677856, 0,
1.19014502772381063096931677856, 2.04968671750905412037886780951, 3.23864990114924671493142556091, 3.89159705681680618594694693660, 4.72976238457494919343474733063, 5.08354147988145076291484449852, 5.93665291599261526039250972586, 6.65701861079484676778234933882, 7.61122226004038614019204616525
