| L(s) = 1 | + 5-s + 5·7-s − 11-s + 6·13-s + 2·17-s − 4·19-s − 7·23-s − 4·25-s + 4·29-s + 10·31-s + 5·35-s + 5·37-s − 11·41-s + 8·43-s − 3·47-s + 18·49-s + 11·53-s − 55-s − 10·59-s + 6·65-s − 10·67-s + 6·71-s − 15·73-s − 5·77-s + 10·79-s − 9·83-s + 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.88·7-s − 0.301·11-s + 1.66·13-s + 0.485·17-s − 0.917·19-s − 1.45·23-s − 4/5·25-s + 0.742·29-s + 1.79·31-s + 0.845·35-s + 0.821·37-s − 1.71·41-s + 1.21·43-s − 0.437·47-s + 18/7·49-s + 1.51·53-s − 0.134·55-s − 1.30·59-s + 0.744·65-s − 1.22·67-s + 0.712·71-s − 1.75·73-s − 0.569·77-s + 1.12·79-s − 0.987·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.706092946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.706092946\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 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.al |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−13.23829877707339, −12.22542076660351, −11.88268516209984, −11.71701964813183, −10.96722602154903, −10.65518249729719, −10.25912329481738, −9.817439186836852, −9.024382304970962, −8.493273213990569, −8.316375623150919, −7.861098082817049, −7.456890331877300, −6.555535745432542, −6.112020871337020, −5.833245953190329, −5.200436546899743, −4.608107782484402, −4.205170532233027, −3.776335850897263, −2.884031282233702, −2.302881804256921, −1.688900903080730, −1.341550196159006, −0.6100209870276387,
0.6100209870276387, 1.341550196159006, 1.688900903080730, 2.302881804256921, 2.884031282233702, 3.776335850897263, 4.205170532233027, 4.608107782484402, 5.200436546899743, 5.833245953190329, 6.112020871337020, 6.555535745432542, 7.456890331877300, 7.861098082817049, 8.316375623150919, 8.493273213990569, 9.024382304970962, 9.817439186836852, 10.25912329481738, 10.65518249729719, 10.96722602154903, 11.71701964813183, 11.88268516209984, 12.22542076660351, 13.23829877707339
