| L(s) = 1 | + 2·5-s + 4·7-s − 6·13-s + 2·17-s + 4·19-s − 4·23-s − 25-s − 6·29-s + 8·35-s − 6·37-s − 6·41-s + 4·43-s + 12·47-s + 9·49-s + 2·53-s + 12·59-s − 14·61-s − 12·65-s − 4·67-s + 12·71-s + 6·73-s + 4·79-s − 4·83-s + 4·85-s − 10·89-s − 24·91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.51·7-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 0.834·23-s − 1/5·25-s − 1.11·29-s + 1.35·35-s − 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 9/7·49-s + 0.274·53-s + 1.56·59-s − 1.79·61-s − 1.48·65-s − 0.488·67-s + 1.42·71-s + 0.702·73-s + 0.450·79-s − 0.439·83-s + 0.433·85-s − 1.05·89-s − 2.51·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−14.17528661553722, −14.00435619130254, −13.68680455863234, −12.85218298001225, −12.23160299299971, −11.96379796896672, −11.48985678370683, −10.76947510518979, −10.37917375515731, −9.784046668403410, −9.418978606037086, −8.893739607254956, −8.001149230214912, −7.863437269748741, −7.199484775984113, −6.769020332415157, −5.717506315630441, −5.436426682750809, −5.141140882411773, −4.349103268162291, −3.827348527710574, −2.860411905366504, −2.217591619249227, −1.807655099762233, −1.117005334360738, 0,
1.117005334360738, 1.807655099762233, 2.217591619249227, 2.860411905366504, 3.827348527710574, 4.349103268162291, 5.141140882411773, 5.436426682750809, 5.717506315630441, 6.769020332415157, 7.199484775984113, 7.863437269748741, 8.001149230214912, 8.893739607254956, 9.418978606037086, 9.784046668403410, 10.37917375515731, 10.76947510518979, 11.48985678370683, 11.96379796896672, 12.23160299299971, 12.85218298001225, 13.68680455863234, 14.00435619130254, 14.17528661553722
