| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 3·11-s + 2·13-s + 2·14-s + 16-s − 19-s + 3·22-s − 6·23-s − 5·25-s − 2·26-s − 2·28-s + 6·29-s + 4·31-s − 32-s + 4·37-s + 38-s + 9·41-s − 43-s − 3·44-s + 6·46-s + 6·47-s − 3·49-s + 5·50-s + 2·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.904·11-s + 0.554·13-s + 0.534·14-s + 1/4·16-s − 0.229·19-s + 0.639·22-s − 1.25·23-s − 25-s − 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.657·37-s + 0.162·38-s + 1.40·41-s − 0.152·43-s − 0.452·44-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46818 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46818 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6792641268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6792641268\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−14.55340177535192, −14.09225095432645, −13.53356917897655, −13.04356674731858, −12.46283556215005, −12.06550960858942, −11.36466896073644, −10.91239520626110, −10.29549447880491, −9.886643593573281, −9.530041589216609, −8.729485225914504, −8.329856881383806, −7.695078198205187, −7.392323166008183, −6.433646163438994, −6.083713463301551, −5.728602642230088, −4.651502114427737, −4.206960228469411, −3.299699963503747, −2.798953541594166, −2.133515876796682, −1.289169597417051, −0.3375450431280792,
0.3375450431280792, 1.289169597417051, 2.133515876796682, 2.798953541594166, 3.299699963503747, 4.206960228469411, 4.651502114427737, 5.728602642230088, 6.083713463301551, 6.433646163438994, 7.392323166008183, 7.695078198205187, 8.329856881383806, 8.729485225914504, 9.530041589216609, 9.886643593573281, 10.29549447880491, 10.91239520626110, 11.36466896073644, 12.06550960858942, 12.46283556215005, 13.04356674731858, 13.53356917897655, 14.09225095432645, 14.55340177535192
