| L(s) = 1 | + 3-s − 5-s − 2·9-s + 5·11-s − 13-s − 15-s − 3·17-s + 5·19-s − 3·23-s − 4·25-s − 5·27-s − 6·29-s + 3·31-s + 5·33-s − 3·37-s − 39-s − 6·41-s − 8·43-s + 2·45-s + 9·47-s − 3·51-s − 5·53-s − 5·55-s + 5·57-s − 59-s − 7·61-s + 65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s + 1.50·11-s − 0.277·13-s − 0.258·15-s − 0.727·17-s + 1.14·19-s − 0.625·23-s − 4/5·25-s − 0.962·27-s − 1.11·29-s + 0.538·31-s + 0.870·33-s − 0.493·37-s − 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.298·45-s + 1.31·47-s − 0.420·51-s − 0.686·53-s − 0.674·55-s + 0.662·57-s − 0.130·59-s − 0.896·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5096 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 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
−7.963219188678038743652738824172, −7.22406177635590176131606667508, −6.50178937599281148964644286638, −5.74190995609850859791287556193, −4.86374468657976715361791849120, −3.83509225624487105163975937390, −3.50954570647175664462717664438, −2.40786173905135505099356644790, −1.47848823003029190038004519897, 0,
1.47848823003029190038004519897, 2.40786173905135505099356644790, 3.50954570647175664462717664438, 3.83509225624487105163975937390, 4.86374468657976715361791849120, 5.74190995609850859791287556193, 6.50178937599281148964644286638, 7.22406177635590176131606667508, 7.963219188678038743652738824172
