| L(s) = 1 | − 2-s + 3-s + 4-s + 2·5-s − 6-s + 4·7-s − 8-s + 9-s − 2·10-s + 12-s − 4·13-s − 4·14-s + 2·15-s + 16-s − 2·17-s − 18-s − 8·19-s + 2·20-s + 4·21-s + 4·23-s − 24-s − 25-s + 4·26-s + 27-s + 4·28-s + 6·29-s − 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 1.51·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 1.10·13-s − 1.06·14-s + 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.872·21-s + 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.755·28-s + 1.11·29-s − 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26862 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26862 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 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
−15.27034604675144, −14.98405920402886, −14.61549664255908, −14.09641620643793, −13.37263329004946, −13.05393970695288, −12.17989905809802, −11.77667838926387, −11.07679013243659, −10.48130903256606, −10.19384902618826, −9.474021389513126, −8.924202734613642, −8.379547661940849, −8.074505138569559, −7.321384104905015, −6.714012078424000, −6.208419010019643, −5.241566130612206, −4.778108936626248, −4.231855259438438, −3.077571413329011, −2.342437951091705, −1.914066493433279, −1.324395240849465, 0,
1.324395240849465, 1.914066493433279, 2.342437951091705, 3.077571413329011, 4.231855259438438, 4.778108936626248, 5.241566130612206, 6.208419010019643, 6.714012078424000, 7.321384104905015, 8.074505138569559, 8.379547661940849, 8.924202734613642, 9.474021389513126, 10.19384902618826, 10.48130903256606, 11.07679013243659, 11.77667838926387, 12.17989905809802, 13.05393970695288, 13.37263329004946, 14.09641620643793, 14.61549664255908, 14.98405920402886, 15.27034604675144
