| L(s) = 1 | + 5-s + 4·7-s + 11-s + 2·13-s − 2·17-s − 4·19-s − 4·23-s + 25-s + 10·29-s + 8·31-s + 4·35-s − 2·37-s − 10·41-s + 12·43-s + 12·47-s + 9·49-s + 6·53-s + 55-s − 12·59-s + 10·61-s + 2·65-s + 4·67-s − 12·71-s − 14·73-s + 4·77-s + 12·79-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.85·29-s + 1.43·31-s + 0.676·35-s − 0.328·37-s − 1.56·41-s + 1.82·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 1.28·61-s + 0.248·65-s + 0.488·67-s − 1.42·71-s − 1.63·73-s + 0.455·77-s + 1.35·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.704225838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.704225838\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−8.537914348852744772961912993584, −7.900671792632468618228665916844, −6.93494649174396324521926238822, −6.22666893956268362722796062173, −5.50580472450737969546590154201, −4.51476337842813499451331647779, −4.19448566901174269835137325397, −2.75719584620147143651296043441, −1.92949333918943451293769928531, −1.01527971888112351803013832105,
1.01527971888112351803013832105, 1.92949333918943451293769928531, 2.75719584620147143651296043441, 4.19448566901174269835137325397, 4.51476337842813499451331647779, 5.50580472450737969546590154201, 6.22666893956268362722796062173, 6.93494649174396324521926238822, 7.900671792632468618228665916844, 8.537914348852744772961912993584
