| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 5-s − 2·6-s + 2·7-s + 9-s − 2·10-s + 5·11-s + 2·12-s − 4·14-s + 15-s − 4·16-s − 4·17-s − 2·18-s + 19-s + 2·20-s + 2·21-s − 10·22-s + 25-s + 27-s + 4·28-s + 2·29-s − 2·30-s + 7·31-s + 8·32-s + 5·33-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 0.447·5-s − 0.816·6-s + 0.755·7-s + 1/3·9-s − 0.632·10-s + 1.50·11-s + 0.577·12-s − 1.06·14-s + 0.258·15-s − 16-s − 0.970·17-s − 0.471·18-s + 0.229·19-s + 0.447·20-s + 0.436·21-s − 2.13·22-s + 1/5·25-s + 0.192·27-s + 0.755·28-s + 0.371·29-s − 0.365·30-s + 1.25·31-s + 1.41·32-s + 0.870·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.734878846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.734878846\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 9 T + p T^{2} \) | 1.61.j |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.116312540328029648976232066022, −7.37658537550803758610520679671, −6.60536436001324492611903123404, −6.21466269871023527137316144132, −4.77489248630681661582881480876, −4.43880894748406576069877086965, −3.31106265407404648860258482457, −2.24300979621442708542358828072, −1.61376707844173736977261500555, −0.853466801303023770515472677809,
0.853466801303023770515472677809, 1.61376707844173736977261500555, 2.24300979621442708542358828072, 3.31106265407404648860258482457, 4.43880894748406576069877086965, 4.77489248630681661582881480876, 6.21466269871023527137316144132, 6.60536436001324492611903123404, 7.37658537550803758610520679671, 8.116312540328029648976232066022
