| L(s) = 1 | + 2·2-s + 2·4-s + 7-s − 6·11-s + 7·13-s + 2·14-s − 4·16-s + 4·17-s + 5·19-s − 12·22-s + 2·23-s + 14·26-s + 2·28-s − 8·29-s + 3·31-s − 8·32-s + 8·34-s − 37-s + 10·38-s + 2·41-s + 11·43-s − 12·44-s + 4·46-s + 4·47-s − 6·49-s + 14·52-s − 16·58-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 1.80·11-s + 1.94·13-s + 0.534·14-s − 16-s + 0.970·17-s + 1.14·19-s − 2.55·22-s + 0.417·23-s + 2.74·26-s + 0.377·28-s − 1.48·29-s + 0.538·31-s − 1.41·32-s + 1.37·34-s − 0.164·37-s + 1.62·38-s + 0.312·41-s + 1.67·43-s − 1.80·44-s + 0.589·46-s + 0.583·47-s − 6/7·49-s + 1.94·52-s − 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.696205625\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.696205625\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−7.71049689281970920478666731408, −7.03849186168759275384764611244, −5.91317864498170255821684595162, −5.69673302677357834035192383602, −5.10716427793505231266843230455, −4.30352389699152757256779837901, −3.45816533354976507417190028791, −3.05469269870795467401246611641, −2.06320335653801770417963169660, −0.869183407003449449854837348527,
0.869183407003449449854837348527, 2.06320335653801770417963169660, 3.05469269870795467401246611641, 3.45816533354976507417190028791, 4.30352389699152757256779837901, 5.10716427793505231266843230455, 5.69673302677357834035192383602, 5.91317864498170255821684595162, 7.03849186168759275384764611244, 7.71049689281970920478666731408
