| L(s) = 1 | − 2-s + 4-s − 2·5-s − 8-s + 2·10-s + 2·11-s + 13-s + 16-s + 3·17-s + 4·19-s − 2·20-s − 2·22-s − 3·23-s − 25-s − 26-s − 7·29-s + 11·31-s − 32-s − 3·34-s − 6·37-s − 4·38-s + 2·40-s − 6·41-s + 43-s + 2·44-s + 3·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s + 0.632·10-s + 0.603·11-s + 0.277·13-s + 1/4·16-s + 0.727·17-s + 0.917·19-s − 0.447·20-s − 0.426·22-s − 0.625·23-s − 1/5·25-s − 0.196·26-s − 1.29·29-s + 1.97·31-s − 0.176·32-s − 0.514·34-s − 0.986·37-s − 0.648·38-s + 0.316·40-s − 0.937·41-s + 0.152·43-s + 0.301·44-s + 0.442·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.051930162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.051930162\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 7 T + p T^{2} \) | 1.29.h |
| 31 | \( 1 - 11 T + p T^{2} \) | 1.31.al |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 11 T + p T^{2} \) | 1.71.al |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.692207491505439003595372266413, −8.149259474632010208639933009664, −7.50081794391536701476109235300, −6.77206025715325106952112089107, −5.92058744841100945848362663445, −4.96231199029695381053928434268, −3.82219360233253453090009490035, −3.27085750102907089905785288191, −1.88945277610592292237661176672, −0.72482106189716605193481872454,
0.72482106189716605193481872454, 1.88945277610592292237661176672, 3.27085750102907089905785288191, 3.82219360233253453090009490035, 4.96231199029695381053928434268, 5.92058744841100945848362663445, 6.77206025715325106952112089107, 7.50081794391536701476109235300, 8.149259474632010208639933009664, 8.692207491505439003595372266413
