| L(s) = 1 | − 3-s − 7-s + 9-s − 2·11-s − 13-s + 4·17-s + 4·19-s + 21-s + 2·23-s − 5·25-s − 27-s + 6·29-s + 2·33-s + 10·37-s + 39-s + 4·41-s − 4·43-s + 49-s − 4·51-s + 10·53-s − 4·57-s + 4·59-s − 2·61-s − 63-s − 2·69-s − 6·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.277·13-s + 0.970·17-s + 0.917·19-s + 0.218·21-s + 0.417·23-s − 25-s − 0.192·27-s + 1.11·29-s + 0.348·33-s + 1.64·37-s + 0.160·39-s + 0.624·41-s − 0.609·43-s + 1/7·49-s − 0.560·51-s + 1.37·53-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.125·63-s − 0.240·69-s − 0.712·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.216016714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.216016714\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−9.955487886234425130452147059205, −9.247206293796622313629696281790, −8.009684812711196026229557536069, −7.43630935606205898492524771367, −6.39863413902985489618747235379, −5.59646731049617673038571558956, −4.82072412949781984303776688240, −3.63231245675015303283666346222, −2.53493343118441382352284955657, −0.876699269022104941037726980213,
0.876699269022104941037726980213, 2.53493343118441382352284955657, 3.63231245675015303283666346222, 4.82072412949781984303776688240, 5.59646731049617673038571558956, 6.39863413902985489618747235379, 7.43630935606205898492524771367, 8.009684812711196026229557536069, 9.247206293796622313629696281790, 9.955487886234425130452147059205
