| L(s) = 1 | − 5-s − 4·7-s − 2·13-s + 6·17-s + 4·23-s + 25-s + 2·29-s − 8·31-s + 4·35-s + 6·37-s + 6·41-s + 12·43-s + 12·47-s + 9·49-s + 10·53-s − 8·59-s − 10·61-s + 2·65-s − 12·67-s − 8·71-s + 10·73-s + 16·79-s − 12·83-s − 6·85-s + 6·89-s + 8·91-s + 18·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 0.554·13-s + 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.82·43-s + 1.75·47-s + 9/7·49-s + 1.37·53-s − 1.04·59-s − 1.28·61-s + 0.248·65-s − 1.46·67-s − 0.949·71-s + 1.17·73-s + 1.80·79-s − 1.31·83-s − 0.650·85-s + 0.635·89-s + 0.838·91-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.177560665\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.177560665\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−9.371673648612747409001951336451, −9.037769434460790287543980197092, −7.59218742689290896826109398615, −7.34348704136551069622443049028, −6.18808196971806754663536089748, −5.54756325101160958989341194873, −4.30735404615915412871147345517, −3.39496965789863498939534704406, −2.63266541859390695587775341566, −0.76610188345715646516141880030,
0.76610188345715646516141880030, 2.63266541859390695587775341566, 3.39496965789863498939534704406, 4.30735404615915412871147345517, 5.54756325101160958989341194873, 6.18808196971806754663536089748, 7.34348704136551069622443049028, 7.59218742689290896826109398615, 9.037769434460790287543980197092, 9.371673648612747409001951336451
