| L(s) = 1 | − 3-s + 9-s − 4·11-s + 13-s + 6·17-s + 6·19-s − 5·25-s − 27-s + 2·29-s + 6·31-s + 4·33-s − 10·37-s − 39-s − 8·41-s − 12·43-s − 12·47-s − 6·51-s + 6·53-s − 6·57-s + 2·61-s − 2·67-s − 8·71-s − 14·73-s + 5·75-s + 4·79-s + 81-s + 8·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.45·17-s + 1.37·19-s − 25-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.696·33-s − 1.64·37-s − 0.160·39-s − 1.24·41-s − 1.82·43-s − 1.75·47-s − 0.840·51-s + 0.824·53-s − 0.794·57-s + 0.256·61-s − 0.244·67-s − 0.949·71-s − 1.63·73-s + 0.577·75-s + 0.450·79-s + 1/9·81-s + 0.878·83-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
| \(L(1)\) |
\(\approx\) |
\(0.9589717595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9589717595\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
| 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
−13.51469316889272, −13.27890968506081, −12.34598740063523, −12.08337170154031, −11.70647922599563, −11.25920605485596, −10.45074692515888, −10.10142723026544, −9.967151292551111, −9.284659135082342, −8.401423878761336, −8.133561699471733, −7.698669552995646, −6.957013699410456, −6.712523740152884, −5.769437170023222, −5.532368591618577, −5.053101668247764, −4.581113227543223, −3.605782021050761, −3.272150573711700, −2.713489083975366, −1.676902980733443, −1.313258927964166, −0.3164990104254057,
0.3164990104254057, 1.313258927964166, 1.676902980733443, 2.713489083975366, 3.272150573711700, 3.605782021050761, 4.581113227543223, 5.053101668247764, 5.532368591618577, 5.769437170023222, 6.712523740152884, 6.957013699410456, 7.698669552995646, 8.133561699471733, 8.401423878761336, 9.284659135082342, 9.967151292551111, 10.10142723026544, 10.45074692515888, 11.25920605485596, 11.70647922599563, 12.08337170154031, 12.34598740063523, 13.27890968506081, 13.51469316889272
