| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 6·11-s + 2·13-s + 14-s + 16-s + 3·17-s + 2·19-s − 6·22-s − 2·26-s − 28-s + 8·31-s − 32-s − 3·34-s + 2·37-s − 2·38-s + 3·41-s − 10·43-s + 6·44-s − 3·47-s − 6·49-s + 2·52-s + 12·53-s + 56-s + 6·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s − 1.27·22-s − 0.392·26-s − 0.188·28-s + 1.43·31-s − 0.176·32-s − 0.514·34-s + 0.328·37-s − 0.324·38-s + 0.468·41-s − 1.52·43-s + 0.904·44-s − 0.437·47-s − 6/7·49-s + 0.277·52-s + 1.64·53-s + 0.133·56-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.620218866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.620218866\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 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.am |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−8.601201076046940639008237162144, −7.78315526519007186922879068219, −6.93985540300127352858512533236, −6.38682625303600140272956298815, −5.75961091111416722490850100823, −4.57885690423164351304582145717, −3.67879513671357649758276667256, −2.98415531300001126600818812846, −1.65052109241719998693478498231, −0.877622893069039740457627492700,
0.877622893069039740457627492700, 1.65052109241719998693478498231, 2.98415531300001126600818812846, 3.67879513671357649758276667256, 4.57885690423164351304582145717, 5.75961091111416722490850100823, 6.38682625303600140272956298815, 6.93985540300127352858512533236, 7.78315526519007186922879068219, 8.601201076046940639008237162144
