| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s − 3·11-s + 3·13-s − 14-s + 16-s + 8·17-s + 3·18-s − 7·19-s + 3·22-s + 23-s − 3·26-s + 28-s + 7·29-s + 10·31-s − 32-s − 8·34-s − 3·36-s − 4·37-s + 7·38-s + 11·41-s − 5·43-s − 3·44-s − 46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s − 0.904·11-s + 0.832·13-s − 0.267·14-s + 1/4·16-s + 1.94·17-s + 0.707·18-s − 1.60·19-s + 0.639·22-s + 0.208·23-s − 0.588·26-s + 0.188·28-s + 1.29·29-s + 1.79·31-s − 0.176·32-s − 1.37·34-s − 1/2·36-s − 0.657·37-s + 1.13·38-s + 1.71·41-s − 0.762·43-s − 0.452·44-s − 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.109560003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109560003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−9.902870692994477031707774336128, −8.726371248408681572518428509425, −8.272057910260049003129414680757, −7.66218664955426905499714394635, −6.38493194631697998032071102429, −5.77234998640983676459369051422, −4.71779735857200140684768121419, −3.31384211194057210126995941937, −2.40848264220543729612005007211, −0.891204744434295621705736593891,
0.891204744434295621705736593891, 2.40848264220543729612005007211, 3.31384211194057210126995941937, 4.71779735857200140684768121419, 5.77234998640983676459369051422, 6.38493194631697998032071102429, 7.66218664955426905499714394635, 8.272057910260049003129414680757, 8.726371248408681572518428509425, 9.902870692994477031707774336128
