| L(s) = 1 | − 2-s + 4-s − 5-s − 7-s − 8-s + 10-s − 3·11-s + 14-s + 16-s + 6·17-s − 2·19-s − 20-s + 3·22-s + 3·23-s + 25-s − 28-s + 6·29-s + 7·31-s − 32-s − 6·34-s + 35-s − 5·37-s + 2·38-s + 40-s + 9·41-s + 2·43-s − 3·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s − 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.267·14-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 0.223·20-s + 0.639·22-s + 0.625·23-s + 1/5·25-s − 0.188·28-s + 1.11·29-s + 1.25·31-s − 0.176·32-s − 1.02·34-s + 0.169·35-s − 0.821·37-s + 0.324·38-s + 0.158·40-s + 1.40·41-s + 0.304·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.759504005\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.759504005\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.74821750960384, −13.06541540991789, −12.55884531892756, −12.26978122222405, −11.76490973057990, −11.06334640513104, −10.75767294298598, −10.14723529635045, −9.874659173089992, −9.279379185605917, −8.666757430387059, −8.183537140229566, −7.745934552344738, −7.430584846772419, −6.549320479363553, −6.365804639771854, −5.599863336361908, −4.950505357830261, −4.581212868514704, −3.510190813011303, −3.286746482179621, −2.569512088806682, −1.978117975848228, −0.8947510570965280, −0.6125456870235015,
0.6125456870235015, 0.8947510570965280, 1.978117975848228, 2.569512088806682, 3.286746482179621, 3.510190813011303, 4.581212868514704, 4.950505357830261, 5.599863336361908, 6.365804639771854, 6.549320479363553, 7.430584846772419, 7.745934552344738, 8.183537140229566, 8.666757430387059, 9.279379185605917, 9.874659173089992, 10.14723529635045, 10.75767294298598, 11.06334640513104, 11.76490973057990, 12.26978122222405, 12.55884531892756, 13.06541540991789, 13.74821750960384
