| L(s) = 1 | + 3·3-s + 5-s + 6·9-s + 11-s + 2·13-s + 3·15-s − 3·17-s − 5·19-s − 4·25-s + 9·27-s − 6·29-s − 31-s + 3·33-s + 5·37-s + 6·39-s − 10·41-s + 4·43-s + 6·45-s + 47-s − 9·51-s + 9·53-s + 55-s − 15·57-s + 3·59-s − 3·61-s + 2·65-s − 11·67-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s + 0.301·11-s + 0.554·13-s + 0.774·15-s − 0.727·17-s − 1.14·19-s − 4/5·25-s + 1.73·27-s − 1.11·29-s − 0.179·31-s + 0.522·33-s + 0.821·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s + 0.894·45-s + 0.145·47-s − 1.26·51-s + 1.23·53-s + 0.134·55-s − 1.98·57-s + 0.390·59-s − 0.384·61-s + 0.248·65-s − 1.34·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.46411987688119, −13.09899923100921, −12.42560816869283, −12.04952110123711, −11.23735080039923, −10.87874344610000, −10.35674098783709, −9.763114955941614, −9.395131003474505, −9.029679248667212, −8.523671773103439, −8.206501167242139, −7.660406539476219, −7.161149374700995, −6.568981799713630, −6.203284961394210, −5.507336210955663, −4.846175656457524, −4.136104002470345, −3.830508959544383, −3.421533243737888, −2.556130211769026, −2.210458197556765, −1.791421446780028, −1.085903384754784, 0,
1.085903384754784, 1.791421446780028, 2.210458197556765, 2.556130211769026, 3.421533243737888, 3.830508959544383, 4.136104002470345, 4.846175656457524, 5.507336210955663, 6.203284961394210, 6.568981799713630, 7.161149374700995, 7.660406539476219, 8.206501167242139, 8.523671773103439, 9.029679248667212, 9.395131003474505, 9.763114955941614, 10.35674098783709, 10.87874344610000, 11.23735080039923, 12.04952110123711, 12.42560816869283, 13.09899923100921, 13.46411987688119
