| L(s) = 1 | − 2-s + 4-s − 5-s − 4·7-s − 8-s − 3·9-s + 10-s − 3·11-s + 13-s + 4·14-s + 16-s + 7·17-s + 3·18-s + 2·19-s − 20-s + 3·22-s − 9·23-s − 4·25-s − 26-s − 4·28-s + 4·29-s − 31-s − 32-s − 7·34-s + 4·35-s − 3·36-s − 2·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s − 9-s + 0.316·10-s − 0.904·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 1.69·17-s + 0.707·18-s + 0.458·19-s − 0.223·20-s + 0.639·22-s − 1.87·23-s − 4/5·25-s − 0.196·26-s − 0.755·28-s + 0.742·29-s − 0.179·31-s − 0.176·32-s − 1.20·34-s + 0.676·35-s − 1/2·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| 53 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.32579811403842, −13.82041873598876, −13.43311630427222, −12.61294154144114, −12.30295964058534, −11.86165078444725, −11.39683346202193, −10.63071092955159, −10.24658824196773, −9.711153315089176, −9.521701005994492, −8.671660826158691, −8.111358455681366, −7.881255563861841, −7.292040020832679, −6.578403851411308, −5.988554418203707, −5.704363698262232, −5.111683737902693, −3.932638333633168, −3.614474329257561, −2.898301688048415, −2.592816369181621, −1.580384336587780, −0.5768152042973063, 0,
0.5768152042973063, 1.580384336587780, 2.592816369181621, 2.898301688048415, 3.614474329257561, 3.932638333633168, 5.111683737902693, 5.704363698262232, 5.988554418203707, 6.578403851411308, 7.292040020832679, 7.881255563861841, 8.111358455681366, 8.671660826158691, 9.521701005994492, 9.711153315089176, 10.24658824196773, 10.63071092955159, 11.39683346202193, 11.86165078444725, 12.30295964058534, 12.61294154144114, 13.43311630427222, 13.82041873598876, 14.32579811403842
