| L(s) = 1 | + 2-s + 4-s + 3·5-s + 5·7-s + 8-s + 3·10-s − 6·11-s − 4·13-s + 5·14-s + 16-s − 3·17-s − 19-s + 3·20-s − 6·22-s + 4·25-s − 4·26-s + 5·28-s + 29-s − 4·31-s + 32-s − 3·34-s + 15·35-s − 37-s − 38-s + 3·40-s + 9·41-s − 7·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s + 1.88·7-s + 0.353·8-s + 0.948·10-s − 1.80·11-s − 1.10·13-s + 1.33·14-s + 1/4·16-s − 0.727·17-s − 0.229·19-s + 0.670·20-s − 1.27·22-s + 4/5·25-s − 0.784·26-s + 0.944·28-s + 0.185·29-s − 0.718·31-s + 0.176·32-s − 0.514·34-s + 2.53·35-s − 0.164·37-s − 0.162·38-s + 0.474·40-s + 1.40·41-s − 1.06·43-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
| \(L(1)\) |
\(\approx\) |
\(2.783221377\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.783221377\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−10.73482166293076509203113078656, −10.36665832788883866939159310360, −9.109497480949641538548778552338, −7.987480521096063799248885737020, −7.30170311636574927756275463221, −5.87958673389096713663595768150, −5.13801159009478652046400048130, −4.61358895125554701935773223253, −2.54786728707166221497736882776, −1.92188031498074361434341547863,
1.92188031498074361434341547863, 2.54786728707166221497736882776, 4.61358895125554701935773223253, 5.13801159009478652046400048130, 5.87958673389096713663595768150, 7.30170311636574927756275463221, 7.987480521096063799248885737020, 9.109497480949641538548778552338, 10.36665832788883866939159310360, 10.73482166293076509203113078656
