| L(s) = 1 | + 3-s − 2·5-s + 9-s − 4·11-s − 13-s − 2·15-s + 6·17-s − 4·19-s − 25-s + 27-s + 4·29-s − 4·31-s − 4·33-s − 12·37-s − 39-s + 12·41-s − 8·43-s − 2·45-s − 2·47-s + 6·51-s − 8·53-s + 8·55-s − 4·57-s − 4·59-s − 10·61-s + 2·65-s − 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.516·15-s + 1.45·17-s − 0.917·19-s − 1/5·25-s + 0.192·27-s + 0.742·29-s − 0.718·31-s − 0.696·33-s − 1.97·37-s − 0.160·39-s + 1.87·41-s − 1.21·43-s − 0.298·45-s − 0.291·47-s + 0.840·51-s − 1.09·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s − 1.28·61-s + 0.248·65-s − 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4379337431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4379337431\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.71709676631732, −12.92369874913207, −12.57521810520626, −12.23270624298952, −11.70538848042876, −11.07224703601751, −10.55147481708504, −10.20470776910065, −9.743780815162113, −8.970530667558775, −8.630761610125394, −7.988030227069726, −7.688836083182574, −7.367245870620344, −6.696315092465356, −5.940737639608639, −5.495199137457413, −4.779362933379735, −4.392302491777516, −3.706125260787177, −3.084761830975998, −2.843786070706114, −1.902196177832624, −1.352466960245073, −0.1893671872973841,
0.1893671872973841, 1.352466960245073, 1.902196177832624, 2.843786070706114, 3.084761830975998, 3.706125260787177, 4.392302491777516, 4.779362933379735, 5.495199137457413, 5.940737639608639, 6.696315092465356, 7.367245870620344, 7.688836083182574, 7.988030227069726, 8.630761610125394, 8.970530667558775, 9.743780815162113, 10.20470776910065, 10.55147481708504, 11.07224703601751, 11.70538848042876, 12.23270624298952, 12.57521810520626, 12.92369874913207, 13.71709676631732
