| L(s) = 1 | − 2-s − 3·3-s + 4-s − 5-s + 3·6-s − 7-s − 8-s + 6·9-s + 10-s + 6·11-s − 3·12-s + 13-s + 14-s + 3·15-s + 16-s + 6·17-s − 6·18-s + 3·19-s − 20-s + 3·21-s − 6·22-s + 3·24-s − 4·25-s − 26-s − 9·27-s − 28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s + 0.316·10-s + 1.80·11-s − 0.866·12-s + 0.277·13-s + 0.267·14-s + 0.774·15-s + 1/4·16-s + 1.45·17-s − 1.41·18-s + 0.688·19-s − 0.223·20-s + 0.654·21-s − 1.27·22-s + 0.612·24-s − 4/5·25-s − 0.196·26-s − 1.73·27-s − 0.188·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39326 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9164366424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9164366424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 53 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.76678385045662, −14.48291985148369, −13.78465968707915, −12.99074234221997, −12.28522246820916, −12.01356416091567, −11.78318474552876, −11.12984482948452, −10.74272302946173, −9.987537541056510, −9.695285134046147, −9.144738905715895, −8.393991567287418, −7.642259043514163, −7.197659425222675, −6.620233568644731, −6.110255462590142, −5.731606359255815, −5.025895424852973, −4.197166586130572, −3.738422450169129, −2.990337460696330, −1.646136699731503, −1.128139462370849, −0.5390876750773419,
0.5390876750773419, 1.128139462370849, 1.646136699731503, 2.990337460696330, 3.738422450169129, 4.197166586130572, 5.025895424852973, 5.731606359255815, 6.110255462590142, 6.620233568644731, 7.197659425222675, 7.642259043514163, 8.393991567287418, 9.144738905715895, 9.695285134046147, 9.987537541056510, 10.74272302946173, 11.12984482948452, 11.78318474552876, 12.01356416091567, 12.28522246820916, 12.99074234221997, 13.78465968707915, 14.48291985148369, 14.76678385045662
