| L(s) = 1 | + 2·2-s + 2·4-s − 3·5-s − 6·10-s − 4·16-s + 2·17-s + 19-s − 6·20-s − 23-s + 4·25-s − 5·29-s + 5·31-s − 8·32-s + 4·34-s − 8·37-s + 2·38-s + 10·41-s − 9·43-s − 2·46-s + 7·47-s + 8·50-s − 9·53-s − 10·58-s − 4·59-s + 8·61-s + 10·62-s − 8·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 1.34·5-s − 1.89·10-s − 16-s + 0.485·17-s + 0.229·19-s − 1.34·20-s − 0.208·23-s + 4/5·25-s − 0.928·29-s + 0.898·31-s − 1.41·32-s + 0.685·34-s − 1.31·37-s + 0.324·38-s + 1.56·41-s − 1.37·43-s − 0.294·46-s + 1.02·47-s + 1.13·50-s − 1.23·53-s − 1.31·58-s − 0.520·59-s + 1.02·61-s + 1.27·62-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.099654548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099654548\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−14.07898189251715, −13.64934355069845, −13.02242367197663, −12.51847280158610, −12.20081723035853, −11.75695280265293, −11.24383932298141, −10.94671156736158, −10.11202598574834, −9.607486936502123, −8.819704291677852, −8.466818018937462, −7.646240280872952, −7.451973002837997, −6.730058359345898, −6.192080910306395, −5.566733913157831, −5.061450444237424, −4.475158622370265, −3.991715787248406, −3.519279076647089, −3.047347415438057, −2.346430518745990, −1.419164067453768, −0.3729346894048540,
0.3729346894048540, 1.419164067453768, 2.346430518745990, 3.047347415438057, 3.519279076647089, 3.991715787248406, 4.475158622370265, 5.061450444237424, 5.566733913157831, 6.192080910306395, 6.730058359345898, 7.451973002837997, 7.646240280872952, 8.466818018937462, 8.819704291677852, 9.607486936502123, 10.11202598574834, 10.94671156736158, 11.24383932298141, 11.75695280265293, 12.20081723035853, 12.51847280158610, 13.02242367197663, 13.64934355069845, 14.07898189251715
