| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 3·11-s + 6·13-s − 15-s + 5·19-s − 21-s + 4·23-s + 25-s − 27-s − 5·29-s + 8·31-s + 3·33-s + 35-s + 7·37-s − 6·39-s − 5·41-s + 45-s − 3·47-s − 6·49-s + 3·53-s − 3·55-s − 5·57-s + 12·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.66·13-s − 0.258·15-s + 1.14·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s + 1.43·31-s + 0.522·33-s + 0.169·35-s + 1.15·37-s − 0.960·39-s − 0.780·41-s + 0.149·45-s − 0.437·47-s − 6/7·49-s + 0.412·53-s − 0.404·55-s − 0.662·57-s + 1.56·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.814341144\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.814341144\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 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
−14.02292285180380, −13.46127375808869, −13.23640074198361, −12.84276592606941, −12.00028091483585, −11.56613426631440, −11.11670766115920, −10.75710685940226, −10.12373772654667, −9.682850955914796, −9.122414043174762, −8.370097157559691, −8.116089681157993, −7.417724564840490, −6.788548205668296, −6.314340793761822, −5.559312091065587, −5.447357420991644, −4.685849350298418, −4.112906720331817, −3.274296325112707, −2.839580397628365, −1.886891167480890, −1.237156464266293, −0.6435312591257324,
0.6435312591257324, 1.237156464266293, 1.886891167480890, 2.839580397628365, 3.274296325112707, 4.112906720331817, 4.685849350298418, 5.447357420991644, 5.559312091065587, 6.314340793761822, 6.788548205668296, 7.417724564840490, 8.116089681157993, 8.370097157559691, 9.122414043174762, 9.682850955914796, 10.12373772654667, 10.75710685940226, 11.11670766115920, 11.56613426631440, 12.00028091483585, 12.84276592606941, 13.23640074198361, 13.46127375808869, 14.02292285180380
