| L(s) = 1 | + 3-s + 5-s − 3·7-s + 9-s − 11-s − 6·13-s + 15-s + 17-s − 19-s − 3·21-s + 6·23-s + 25-s + 27-s − 5·29-s − 2·31-s − 33-s − 3·35-s − 11·37-s − 6·39-s − 9·41-s + 45-s + 5·47-s + 2·49-s + 51-s − 3·53-s − 55-s − 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.258·15-s + 0.242·17-s − 0.229·19-s − 0.654·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s − 0.928·29-s − 0.359·31-s − 0.174·33-s − 0.507·35-s − 1.80·37-s − 0.960·39-s − 1.40·41-s + 0.149·45-s + 0.729·47-s + 2/7·49-s + 0.140·51-s − 0.412·53-s − 0.134·55-s − 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.115995186356517566132049383843, −7.88838720410263343605234437105, −7.15203562927167970765525330914, −6.58191745816098313511312960641, −5.45751654674526263968548276270, −4.77678464018513992595769282811, −3.47953954157197698189421883851, −2.84769201911839488867343006498, −1.84045839091382893383997095118, 0,
1.84045839091382893383997095118, 2.84769201911839488867343006498, 3.47953954157197698189421883851, 4.77678464018513992595769282811, 5.45751654674526263968548276270, 6.58191745816098313511312960641, 7.15203562927167970765525330914, 7.88838720410263343605234437105, 9.115995186356517566132049383843
