| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s + 6·13-s + 2·15-s + 6·17-s + 4·19-s + 2·21-s − 2·23-s − 25-s − 27-s + 6·29-s − 8·31-s + 4·35-s + 8·37-s − 6·39-s + 6·41-s + 43-s − 2·45-s − 2·47-s − 3·49-s − 6·51-s − 12·53-s − 4·57-s + 8·59-s − 8·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s + 1.66·13-s + 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.436·21-s − 0.417·23-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s + 1.31·37-s − 0.960·39-s + 0.937·41-s + 0.152·43-s − 0.298·45-s − 0.291·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 0.529·57-s + 1.04·59-s − 1.02·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.323547698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.323547698\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−7.84361416221708167270476836121, −7.15093889058481373399329505929, −6.26298458146440518268581453940, −5.89793255006229174859407262053, −5.10371010153033600597674759928, −4.08317912738990129898482139403, −3.59412705396114273825373123773, −2.95618601672140055443950487370, −1.44700622859980288853212475386, −0.63556009529059807189390205252,
0.63556009529059807189390205252, 1.44700622859980288853212475386, 2.95618601672140055443950487370, 3.59412705396114273825373123773, 4.08317912738990129898482139403, 5.10371010153033600597674759928, 5.89793255006229174859407262053, 6.26298458146440518268581453940, 7.15093889058481373399329505929, 7.84361416221708167270476836121
