| L(s) = 1 | + 5-s + 4·7-s − 6·11-s − 3·17-s + 7·19-s − 9·23-s + 25-s + 7·31-s + 4·35-s − 2·37-s − 6·41-s + 2·43-s + 9·49-s + 9·53-s − 6·55-s + 12·59-s − 7·61-s − 2·67-s − 6·71-s − 2·73-s − 24·77-s − 79-s − 9·83-s − 3·85-s + 6·89-s + 7·95-s − 8·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.51·7-s − 1.80·11-s − 0.727·17-s + 1.60·19-s − 1.87·23-s + 1/5·25-s + 1.25·31-s + 0.676·35-s − 0.328·37-s − 0.937·41-s + 0.304·43-s + 9/7·49-s + 1.23·53-s − 0.809·55-s + 1.56·59-s − 0.896·61-s − 0.244·67-s − 0.712·71-s − 0.234·73-s − 2.73·77-s − 0.112·79-s − 0.987·83-s − 0.325·85-s + 0.635·89-s + 0.718·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.500528424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.500528424\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.81116288850840, −13.49812316564369, −13.04879969313031, −12.22756648950085, −11.81090992592377, −11.50908532275833, −10.81154790390380, −10.32899795036895, −10.06978624107766, −9.463161815853171, −8.607265987025469, −8.335992357932636, −7.828757808557493, −7.439117098867607, −6.833325846706210, −5.978238715327372, −5.503219469957109, −5.160980820588912, −4.586540449870232, −4.068820935291837, −3.131145917333215, −2.525261674865841, −2.029652794482271, −1.397484210742555, −0.4954394215140356,
0.4954394215140356, 1.397484210742555, 2.029652794482271, 2.525261674865841, 3.131145917333215, 4.068820935291837, 4.586540449870232, 5.160980820588912, 5.503219469957109, 5.978238715327372, 6.833325846706210, 7.439117098867607, 7.828757808557493, 8.335992357932636, 8.607265987025469, 9.463161815853171, 10.06978624107766, 10.32899795036895, 10.81154790390380, 11.50908532275833, 11.81090992592377, 12.22756648950085, 13.04879969313031, 13.49812316564369, 13.81116288850840
