| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s + 11-s + 2·13-s − 15-s + 4·21-s + 9·23-s − 4·25-s − 27-s + 4·29-s − 5·31-s − 33-s − 4·35-s + 10·37-s − 2·39-s − 4·41-s + 6·43-s + 45-s + 47-s + 9·49-s + 2·53-s + 55-s − 12·59-s − 4·63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.258·15-s + 0.872·21-s + 1.87·23-s − 4/5·25-s − 0.192·27-s + 0.742·29-s − 0.898·31-s − 0.174·33-s − 0.676·35-s + 1.64·37-s − 0.320·39-s − 0.624·41-s + 0.914·43-s + 0.149·45-s + 0.145·47-s + 9/7·49-s + 0.274·53-s + 0.134·55-s − 1.56·59-s − 0.503·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
| 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
−13.51871418272966, −13.06931147885201, −12.65784351634226, −12.22098755082197, −11.69744515563877, −11.06666413692838, −10.74005826279155, −10.25340361558647, −9.656723455633196, −9.235386536945076, −9.066247696503709, −8.285880111797098, −7.501784625714237, −7.192352918328922, −6.517821685109475, −6.063391022920810, −6.002135136345856, −5.099856268506927, −4.690800206557877, −3.877348783770159, −3.474742489098766, −2.852117569410229, −2.293572994318907, −1.346317431258522, −0.8287178449727613, 0,
0.8287178449727613, 1.346317431258522, 2.293572994318907, 2.852117569410229, 3.474742489098766, 3.877348783770159, 4.690800206557877, 5.099856268506927, 6.002135136345856, 6.063391022920810, 6.517821685109475, 7.192352918328922, 7.501784625714237, 8.285880111797098, 9.066247696503709, 9.235386536945076, 9.656723455633196, 10.25340361558647, 10.74005826279155, 11.06666413692838, 11.69744515563877, 12.22098755082197, 12.65784351634226, 13.06931147885201, 13.51871418272966
