| L(s) = 1 | − 2-s − 4-s − 2·7-s + 3·8-s + 5·11-s − 4·13-s + 2·14-s − 16-s − 2·17-s − 19-s − 5·22-s − 8·23-s − 5·25-s + 4·26-s + 2·28-s − 29-s − 3·31-s − 5·32-s + 2·34-s + 38-s + 3·41-s − 10·43-s − 5·44-s + 8·46-s + 9·47-s − 3·49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.755·7-s + 1.06·8-s + 1.50·11-s − 1.10·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s − 0.229·19-s − 1.06·22-s − 1.66·23-s − 25-s + 0.784·26-s + 0.377·28-s − 0.185·29-s − 0.538·31-s − 0.883·32-s + 0.342·34-s + 0.162·38-s + 0.468·41-s − 1.52·43-s − 0.753·44-s + 1.17·46-s + 1.31·47-s − 3/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 + 13 T + p T^{2} \) | 1.53.n |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−9.975068001100058974663401806842, −9.664464461389608067230478328697, −8.852294078640656732090499635948, −7.87417052040465255352940163624, −6.91882834173448692854531317411, −5.94744779919123652206609870128, −4.53756817363146772269371926226, −3.68960087010757701654343365818, −1.87623922290012157749248556202, 0,
1.87623922290012157749248556202, 3.68960087010757701654343365818, 4.53756817363146772269371926226, 5.94744779919123652206609870128, 6.91882834173448692854531317411, 7.87417052040465255352940163624, 8.852294078640656732090499635948, 9.664464461389608067230478328697, 9.975068001100058974663401806842
