| L(s) = 1 | − 2-s + 4-s + 3·5-s + 7-s − 8-s − 3·10-s + 3·11-s + 5·13-s − 14-s + 16-s − 17-s + 2·19-s + 3·20-s − 3·22-s + 4·25-s − 5·26-s + 28-s + 2·31-s − 32-s + 34-s + 3·35-s − 37-s − 2·38-s − 3·40-s + 5·43-s + 3·44-s − 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s + 0.377·7-s − 0.353·8-s − 0.948·10-s + 0.904·11-s + 1.38·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.458·19-s + 0.670·20-s − 0.639·22-s + 4/5·25-s − 0.980·26-s + 0.188·28-s + 0.359·31-s − 0.176·32-s + 0.171·34-s + 0.507·35-s − 0.164·37-s − 0.324·38-s − 0.474·40-s + 0.762·43-s + 0.452·44-s − 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.015538787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.015538787\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−9.052138670987373994541813813241, −8.583687013397882071985645325670, −7.62211801412416468643283924364, −6.59038139553706762277955973954, −6.13784106971061118098727945401, −5.34980418823681217755725401346, −4.15556988809290758674718549938, −3.02852706937252409484105898746, −1.84054601317722139227417509529, −1.17615148289952457683042205285,
1.17615148289952457683042205285, 1.84054601317722139227417509529, 3.02852706937252409484105898746, 4.15556988809290758674718549938, 5.34980418823681217755725401346, 6.13784106971061118098727945401, 6.59038139553706762277955973954, 7.62211801412416468643283924364, 8.583687013397882071985645325670, 9.052138670987373994541813813241
