| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 2·11-s + 4·13-s − 2·14-s + 16-s − 17-s + 4·19-s + 20-s + 2·22-s + 3·23-s − 4·25-s − 4·26-s + 2·28-s + 2·29-s + 4·31-s − 32-s + 34-s + 2·35-s − 37-s − 4·38-s − 40-s + 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.603·11-s + 1.10·13-s − 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.223·20-s + 0.426·22-s + 0.625·23-s − 4/5·25-s − 0.784·26-s + 0.377·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.164·37-s − 0.648·38-s − 0.158·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.431111098\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.431111098\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 61 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−9.835228703765265785943105604317, −9.007866832147803798553086276195, −8.237018475688922551029648913739, −7.60245567784740990417597755627, −6.55591130528069352896819785570, −5.69728244342077135331317273685, −4.79565635056518088996323913879, −3.43918150087291676647094176137, −2.22874177276732534348962813046, −1.08643652995918499542816592514,
1.08643652995918499542816592514, 2.22874177276732534348962813046, 3.43918150087291676647094176137, 4.79565635056518088996323913879, 5.69728244342077135331317273685, 6.55591130528069352896819785570, 7.60245567784740990417597755627, 8.237018475688922551029648913739, 9.007866832147803798553086276195, 9.835228703765265785943105604317
