| L(s) = 1 | + 2-s + 4-s + 4·5-s + 8-s + 4·10-s − 4·11-s + 13-s + 16-s − 2·17-s − 4·19-s + 4·20-s − 4·22-s + 2·23-s + 11·25-s + 26-s + 6·29-s + 4·31-s + 32-s − 2·34-s + 4·37-s − 4·38-s + 4·40-s + 6·41-s − 4·43-s − 4·44-s + 2·46-s + 11·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.353·8-s + 1.26·10-s − 1.20·11-s + 0.277·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.894·20-s − 0.852·22-s + 0.417·23-s + 11/5·25-s + 0.196·26-s + 1.11·29-s + 0.718·31-s + 0.176·32-s − 0.342·34-s + 0.657·37-s − 0.648·38-s + 0.632·40-s + 0.937·41-s − 0.609·43-s − 0.603·44-s + 0.294·46-s + 1.55·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.931757583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.931757583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.54655634536391, −15.67761689825292, −15.35581921421084, −14.47847130127235, −14.18224893546657, −13.42795668772182, −13.12153675893778, −12.81350119289852, −11.97658491800633, −11.14410057698489, −10.55572888996987, −10.21328098073293, −9.581293071031431, −8.785620557395911, −8.266246901983622, −7.350065836976429, −6.532362776235923, −6.191995792606228, −5.527216069886748, −4.919498095453959, −4.356192221884600, −3.143409183534610, −2.480612285259507, −2.035995548126316, −0.9198558327568641,
0.9198558327568641, 2.035995548126316, 2.480612285259507, 3.143409183534610, 4.356192221884600, 4.919498095453959, 5.527216069886748, 6.191995792606228, 6.532362776235923, 7.350065836976429, 8.266246901983622, 8.785620557395911, 9.581293071031431, 10.21328098073293, 10.55572888996987, 11.14410057698489, 11.97658491800633, 12.81350119289852, 13.12153675893778, 13.42795668772182, 14.18224893546657, 14.47847130127235, 15.35581921421084, 15.67761689825292, 16.54655634536391
