| L(s) = 1 | + 3-s + 2·5-s + 9-s − 11-s − 4·13-s + 2·15-s + 6·19-s − 4·23-s − 25-s + 27-s + 8·29-s − 33-s + 37-s − 4·39-s − 6·41-s + 2·43-s + 2·45-s − 8·47-s − 7·49-s − 6·53-s − 2·55-s + 6·57-s − 12·59-s − 4·61-s − 8·65-s − 4·67-s − 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.516·15-s + 1.37·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s + 1.48·29-s − 0.174·33-s + 0.164·37-s − 0.640·39-s − 0.937·41-s + 0.304·43-s + 0.298·45-s − 1.16·47-s − 49-s − 0.824·53-s − 0.269·55-s + 0.794·57-s − 1.56·59-s − 0.512·61-s − 0.992·65-s − 0.488·67-s − 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 37 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−15.96469783457638, −15.43381985777317, −14.71831282690931, −14.26759206244375, −13.80392215824291, −13.43259713027828, −12.74193672902056, −12.11228632157421, −11.74911532376318, −10.87309665033724, −10.10808873548988, −9.854666395868772, −9.429533085109384, −8.700953892288329, −7.941122427418378, −7.622027694619287, −6.816613728753039, −6.222258897236007, −5.520354208439762, −4.877880388168336, −4.348403573599934, −3.170706417923863, −2.886764473414258, −1.966574787003098, −1.364688516184232, 0,
1.364688516184232, 1.966574787003098, 2.886764473414258, 3.170706417923863, 4.348403573599934, 4.877880388168336, 5.520354208439762, 6.222258897236007, 6.816613728753039, 7.622027694619287, 7.941122427418378, 8.700953892288329, 9.429533085109384, 9.854666395868772, 10.10808873548988, 10.87309665033724, 11.74911532376318, 12.11228632157421, 12.74193672902056, 13.43259713027828, 13.80392215824291, 14.26759206244375, 14.71831282690931, 15.43381985777317, 15.96469783457638
