| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s − 2·9-s − 2·10-s + 11-s − 12-s + 13-s − 2·15-s + 16-s − 4·17-s + 2·18-s − 4·19-s + 2·20-s − 22-s − 5·23-s + 24-s − 25-s − 26-s + 5·27-s + 6·29-s + 2·30-s − 11·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s − 0.970·17-s + 0.471·18-s − 0.917·19-s + 0.447·20-s − 0.213·22-s − 1.04·23-s + 0.204·24-s − 1/5·25-s − 0.196·26-s + 0.962·27-s + 1.11·29-s + 0.365·30-s − 1.97·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 11 T + p T^{2} \) | 1.31.l |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−9.219955275192213099969260456162, −8.658993015233193813263697068074, −7.74743301941803870846936002176, −6.49433799380884794638861677672, −6.19868577061426635646185050789, −5.29483134041303464053290762791, −4.08472177649791044391477953621, −2.63743149324984044028432894824, −1.67425462808243907178495844610, 0,
1.67425462808243907178495844610, 2.63743149324984044028432894824, 4.08472177649791044391477953621, 5.29483134041303464053290762791, 6.19868577061426635646185050789, 6.49433799380884794638861677672, 7.74743301941803870846936002176, 8.658993015233193813263697068074, 9.219955275192213099969260456162
