| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 11-s + 2·13-s + 2·14-s + 16-s + 5·19-s − 22-s − 3·23-s − 5·25-s + 2·26-s + 2·28-s + 9·29-s + 5·31-s + 32-s + 8·37-s + 5·38-s − 6·41-s − 43-s − 44-s − 3·46-s + 3·47-s − 3·49-s − 5·50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.301·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s + 1.14·19-s − 0.213·22-s − 0.625·23-s − 25-s + 0.392·26-s + 0.377·28-s + 1.67·29-s + 0.898·31-s + 0.176·32-s + 1.31·37-s + 0.811·38-s − 0.937·41-s − 0.152·43-s − 0.150·44-s − 0.442·46-s + 0.437·47-s − 3/7·49-s − 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.104091508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.104091508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.364230372081750695095905189873, −8.104490667080905764941414747978, −7.925510322211287368472840649607, −6.73283167278283317174745745748, −5.99222630944204320962138199012, −5.12194653895897539127544078353, −4.41959229383503607898562127552, −3.42733523473514779142644194817, −2.40924789423485261274608244532, −1.19619323141889946587904831006,
1.19619323141889946587904831006, 2.40924789423485261274608244532, 3.42733523473514779142644194817, 4.41959229383503607898562127552, 5.12194653895897539127544078353, 5.99222630944204320962138199012, 6.73283167278283317174745745748, 7.925510322211287368472840649607, 8.104490667080905764941414747978, 9.364230372081750695095905189873
