| L(s) = 1 | + 2-s − 4-s − 5-s − 2·7-s − 3·8-s − 10-s − 5·13-s − 2·14-s − 16-s − 17-s − 19-s + 20-s − 23-s − 4·25-s − 5·26-s + 2·28-s + 9·29-s − 8·31-s + 5·32-s − 34-s + 2·35-s − 2·37-s − 38-s + 3·40-s − 3·41-s + 7·43-s − 46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s − 0.755·7-s − 1.06·8-s − 0.316·10-s − 1.38·13-s − 0.534·14-s − 1/4·16-s − 0.242·17-s − 0.229·19-s + 0.223·20-s − 0.208·23-s − 4/5·25-s − 0.980·26-s + 0.377·28-s + 1.67·29-s − 1.43·31-s + 0.883·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s − 0.162·38-s + 0.474·40-s − 0.468·41-s + 1.06·43-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 11 T + p T^{2} \) | 1.71.l |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−10.57164856619779701457139584281, −9.665629704459847101858554911308, −8.936478055819154740374092802433, −7.80543774392447436966310628509, −6.77348244801183470962935816882, −5.71148956667906146005572789841, −4.69578621390901362334886010759, −3.79485333903042345980929947012, −2.65692004333533929752956723554, 0,
2.65692004333533929752956723554, 3.79485333903042345980929947012, 4.69578621390901362334886010759, 5.71148956667906146005572789841, 6.77348244801183470962935816882, 7.80543774392447436966310628509, 8.936478055819154740374092802433, 9.665629704459847101858554911308, 10.57164856619779701457139584281
