| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s − 4·13-s − 6·17-s − 6·19-s + 21-s − 8·23-s − 27-s − 2·29-s − 10·31-s + 2·33-s − 2·37-s + 4·39-s + 10·41-s − 4·43-s − 8·47-s + 49-s + 6·51-s − 4·53-s + 6·57-s + 8·59-s + 6·61-s − 63-s + 12·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s − 1.45·17-s − 1.37·19-s + 0.218·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s − 1.79·31-s + 0.348·33-s − 0.328·37-s + 0.640·39-s + 1.56·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s + 0.794·57-s + 1.04·59-s + 0.768·61-s − 0.125·63-s + 1.46·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2743079172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2743079172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| 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
−7.78658425695217681289174974487, −6.91769210168230530119950486753, −6.52255862189646900458465165687, −5.69891040102091247679907360501, −5.06763303548809211709491905402, −4.28492920944489376558236890239, −3.68706754584159092001777611990, −2.33362332529370681569023240079, −2.03385004077130300657048608042, −0.24338728321867744055440527565,
0.24338728321867744055440527565, 2.03385004077130300657048608042, 2.33362332529370681569023240079, 3.68706754584159092001777611990, 4.28492920944489376558236890239, 5.06763303548809211709491905402, 5.69891040102091247679907360501, 6.52255862189646900458465165687, 6.91769210168230530119950486753, 7.78658425695217681289174974487
