| L(s) = 1 | − 5-s − 2·7-s − 4·11-s + 13-s + 4·17-s + 6·19-s + 25-s − 4·29-s − 6·31-s + 2·35-s + 2·37-s − 10·41-s + 8·43-s − 3·49-s − 4·53-s + 4·55-s − 4·59-s − 2·61-s − 65-s + 6·67-s − 8·71-s + 10·73-s + 8·77-s + 4·79-s + 12·83-s − 4·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 1.20·11-s + 0.277·13-s + 0.970·17-s + 1.37·19-s + 1/5·25-s − 0.742·29-s − 1.07·31-s + 0.338·35-s + 0.328·37-s − 1.56·41-s + 1.21·43-s − 3/7·49-s − 0.549·53-s + 0.539·55-s − 0.520·59-s − 0.256·61-s − 0.124·65-s + 0.733·67-s − 0.949·71-s + 1.17·73-s + 0.911·77-s + 0.450·79-s + 1.31·83-s − 0.433·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−15.27624912414912, −14.60322997648782, −14.05486282402423, −13.50435149516213, −13.02505141150116, −12.51919337563641, −12.10219107562391, −11.39816815745281, −10.94569319945051, −10.36311207511576, −9.772559963171565, −9.410378523269572, −8.724737172656726, −7.975103059200035, −7.598947533147457, −7.185389228292257, −6.370466265038035, −5.731812902474915, −5.253911261065782, −4.695843551780921, −3.571281559229351, −3.448467652045257, −2.724177097899569, −1.827474369653490, −0.8588430876592652, 0,
0.8588430876592652, 1.827474369653490, 2.724177097899569, 3.448467652045257, 3.571281559229351, 4.695843551780921, 5.253911261065782, 5.731812902474915, 6.370466265038035, 7.185389228292257, 7.598947533147457, 7.975103059200035, 8.724737172656726, 9.410378523269572, 9.772559963171565, 10.36311207511576, 10.94569319945051, 11.39816815745281, 12.10219107562391, 12.51919337563641, 13.02505141150116, 13.50435149516213, 14.05486282402423, 14.60322997648782, 15.27624912414912
