| L(s) = 1 | − 2·5-s − 4·7-s − 4·11-s + 6·13-s − 17-s + 4·19-s + 4·23-s − 25-s + 6·29-s − 4·31-s + 8·35-s + 10·37-s + 6·41-s + 4·43-s + 8·47-s + 9·49-s − 6·53-s + 8·55-s + 4·59-s − 14·61-s − 12·65-s − 12·67-s + 12·71-s + 10·73-s + 16·77-s − 4·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s − 1.20·11-s + 1.66·13-s − 0.242·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.35·35-s + 1.64·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s + 1.07·55-s + 0.520·59-s − 1.79·61-s − 1.48·65-s − 1.46·67-s + 1.42·71-s + 1.17·73-s + 1.82·77-s − 0.450·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.024964484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.024964484\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−9.633532707778361049184968533027, −8.970424755908821973249850855092, −8.003753229013609592486800840857, −7.35949009666835853584688655692, −6.34536531717589705924279835378, −5.68532009157392283288127569559, −4.39115436224503183143289880455, −3.45432480842183618626147488920, −2.78738487231981523171002916538, −0.73884361659554928062200855822,
0.73884361659554928062200855822, 2.78738487231981523171002916538, 3.45432480842183618626147488920, 4.39115436224503183143289880455, 5.68532009157392283288127569559, 6.34536531717589705924279835378, 7.35949009666835853584688655692, 8.003753229013609592486800840857, 8.970424755908821973249850855092, 9.633532707778361049184968533027
