| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s − 12-s − 14-s + 16-s − 17-s − 18-s − 21-s − 6·23-s + 24-s − 5·25-s − 27-s + 28-s + 6·29-s + 6·31-s − 32-s + 34-s + 36-s + 4·37-s + 10·41-s + 42-s + 4·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.218·21-s − 1.25·23-s + 0.204·24-s − 25-s − 0.192·27-s + 0.188·28-s + 1.11·29-s + 1.07·31-s − 0.176·32-s + 0.171·34-s + 1/6·36-s + 0.657·37-s + 1.56·41-s + 0.154·42-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257754 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.06385796071402, −12.33709740289129, −11.95913915685931, −11.75403892018648, −11.12371117122447, −10.78144345705871, −10.21405918934015, −9.894406702440718, −9.440180515475772, −8.878950245323618, −8.288113916018521, −7.958324690118411, −7.510844396810060, −7.000062563010797, −6.241476382314797, −6.123535742261219, −5.610350892338309, −4.797452405995836, −4.443008488384649, −3.918566875015730, −3.181828420259292, −2.437526439541403, −2.084748482744604, −1.275295840740173, −0.7608065867729209, 0,
0.7608065867729209, 1.275295840740173, 2.084748482744604, 2.437526439541403, 3.181828420259292, 3.918566875015730, 4.443008488384649, 4.797452405995836, 5.610350892338309, 6.123535742261219, 6.241476382314797, 7.000062563010797, 7.510844396810060, 7.958324690118411, 8.288113916018521, 8.878950245323618, 9.440180515475772, 9.894406702440718, 10.21405918934015, 10.78144345705871, 11.12371117122447, 11.75403892018648, 11.95913915685931, 12.33709740289129, 13.06385796071402
