| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s + 4·11-s + 2·13-s + 2·15-s − 4·19-s + 21-s − 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s − 4·33-s + 2·35-s − 6·37-s − 2·39-s + 10·41-s + 4·43-s − 2·45-s + 8·47-s + 49-s − 6·53-s − 8·55-s + 4·57-s + 8·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s − 0.917·19-s + 0.218·21-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s + 0.338·35-s − 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.298·45-s + 1.16·47-s + 1/7·49-s − 0.824·53-s − 1.07·55-s + 0.529·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5532408624\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5532408624\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
| 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
−13.12444654309197, −12.32410928842503, −12.28062832332474, −11.64567863632791, −11.29231780319520, −10.83963596788135, −10.40707584808952, −9.754177645772916, −9.327589043882500, −8.768530480255633, −8.442317292456890, −7.658198622868593, −7.334563419287416, −6.899754449716451, −6.121605154560554, −5.933843406431332, −5.488205036761500, −4.370297339846405, −4.280428299708039, −3.670067592337243, −3.429520659820686, −2.257333763680045, −1.855799105686818, −1.052939462042110, −0.2443841967198096,
0.2443841967198096, 1.052939462042110, 1.855799105686818, 2.257333763680045, 3.429520659820686, 3.670067592337243, 4.280428299708039, 4.370297339846405, 5.488205036761500, 5.933843406431332, 6.121605154560554, 6.899754449716451, 7.334563419287416, 7.658198622868593, 8.442317292456890, 8.768530480255633, 9.327589043882500, 9.754177645772916, 10.40707584808952, 10.83963596788135, 11.29231780319520, 11.64567863632791, 12.28062832332474, 12.32410928842503, 13.12444654309197
