| L(s) = 1 | − 3-s − 7-s + 9-s + 3·11-s − 13-s − 17-s − 3·19-s + 21-s + 2·23-s − 27-s + 29-s − 10·31-s − 3·33-s − 7·37-s + 39-s + 5·41-s + 12·43-s + 9·47-s − 6·49-s + 51-s + 11·53-s + 3·57-s − 12·59-s − 2·61-s − 63-s − 2·69-s − 4·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.242·17-s − 0.688·19-s + 0.218·21-s + 0.417·23-s − 0.192·27-s + 0.185·29-s − 1.79·31-s − 0.522·33-s − 1.15·37-s + 0.160·39-s + 0.780·41-s + 1.82·43-s + 1.31·47-s − 6/7·49-s + 0.140·51-s + 1.51·53-s + 0.397·57-s − 1.56·59-s − 0.256·61-s − 0.125·63-s − 0.240·69-s − 0.474·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366095589\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366095589\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.08329024153051, −13.82328611370234, −13.05868865784179, −12.50046063620232, −12.36299013949685, −11.73714165603875, −11.00787512841760, −10.80072964516582, −10.30276138133084, −9.427454207420460, −9.201122680914480, −8.769751061279034, −7.913129443034162, −7.310425250703258, −6.938989527697873, −6.305544897616813, −5.878305189337729, −5.260998446341674, −4.643035523679073, −3.941053957340978, −3.637389873124742, −2.659149598566052, −2.035000261716967, −1.243415606010393, −0.4313133965735780,
0.4313133965735780, 1.243415606010393, 2.035000261716967, 2.659149598566052, 3.637389873124742, 3.941053957340978, 4.643035523679073, 5.260998446341674, 5.878305189337729, 6.305544897616813, 6.938989527697873, 7.310425250703258, 7.913129443034162, 8.769751061279034, 9.201122680914480, 9.427454207420460, 10.30276138133084, 10.80072964516582, 11.00787512841760, 11.73714165603875, 12.36299013949685, 12.50046063620232, 13.05868865784179, 13.82328611370234, 14.08329024153051
