| L(s) = 1 | + 5-s + 2·7-s − 13-s + 3·17-s + 4·19-s + 23-s + 25-s − 4·29-s − 7·31-s + 2·35-s − 2·37-s − 4·41-s + 8·43-s − 3·47-s − 3·49-s − 11·53-s − 6·59-s + 7·61-s − 65-s − 2·67-s − 12·71-s + 2·73-s − 15·79-s + 4·83-s + 3·85-s − 10·89-s − 2·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.208·23-s + 1/5·25-s − 0.742·29-s − 1.25·31-s + 0.338·35-s − 0.328·37-s − 0.624·41-s + 1.21·43-s − 0.437·47-s − 3/7·49-s − 1.51·53-s − 0.781·59-s + 0.896·61-s − 0.124·65-s − 0.244·67-s − 1.42·71-s + 0.234·73-s − 1.68·79-s + 0.439·83-s + 0.325·85-s − 1.05·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
| 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
−12.92802976769124, −12.58238867024804, −12.01188370264409, −11.55087042219143, −11.15331505317711, −10.75965694942858, −10.13166528257383, −9.774834382690844, −9.292907926037530, −8.873339934152585, −8.316832822365790, −7.787905153583814, −7.301590873250418, −7.128980325236204, −6.233968525549756, −5.790490478256801, −5.427561149933320, −4.821571798119977, −4.531795904216840, −3.645676691372149, −3.279464930037938, −2.695149099130066, −1.832620028645277, −1.630587863030656, −0.8774323562178688, 0,
0.8774323562178688, 1.630587863030656, 1.832620028645277, 2.695149099130066, 3.279464930037938, 3.645676691372149, 4.531795904216840, 4.821571798119977, 5.427561149933320, 5.790490478256801, 6.233968525549756, 7.128980325236204, 7.301590873250418, 7.787905153583814, 8.316832822365790, 8.873339934152585, 9.292907926037530, 9.774834382690844, 10.13166528257383, 10.75965694942858, 11.15331505317711, 11.55087042219143, 12.01188370264409, 12.58238867024804, 12.92802976769124
