| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s + 2·13-s − 14-s + 16-s − 3·17-s + 5·19-s + 3·20-s − 6·23-s + 4·25-s + 2·26-s − 28-s − 29-s − 4·31-s + 32-s − 3·34-s − 3·35-s + 11·37-s + 5·38-s + 3·40-s − 3·41-s − 43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.392·26-s − 0.188·28-s − 0.185·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s − 0.507·35-s + 1.80·37-s + 0.811·38-s + 0.474·40-s − 0.468·41-s − 0.152·43-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.597579140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.597579140\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−10.93418497363936375371538356745, −9.903097505682708174047000084797, −9.393317296221992897621160669310, −8.138832094489574982377060977363, −6.91113774260761251105300471991, −6.05605953619251861228079704301, −5.45136901549972449852056937651, −4.16733311374824074298313875215, −2.89342452075666737621670076479, −1.70889636297831516324340078732,
1.70889636297831516324340078732, 2.89342452075666737621670076479, 4.16733311374824074298313875215, 5.45136901549972449852056937651, 6.05605953619251861228079704301, 6.91113774260761251105300471991, 8.138832094489574982377060977363, 9.393317296221992897621160669310, 9.903097505682708174047000084797, 10.93418497363936375371538356745
