| L(s) = 1 | − 3·5-s − 3·7-s − 3·9-s − 4·13-s + 17-s + 3·19-s + 8·23-s + 4·25-s + 2·29-s − 3·31-s + 9·35-s + 3·37-s + 5·43-s + 9·45-s − 6·47-s + 2·49-s − 6·53-s + 3·59-s + 5·61-s + 9·63-s + 12·65-s + 4·67-s + 16·71-s + 4·73-s + 79-s + 9·81-s + 3·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 9-s − 1.10·13-s + 0.242·17-s + 0.688·19-s + 1.66·23-s + 4/5·25-s + 0.371·29-s − 0.538·31-s + 1.52·35-s + 0.493·37-s + 0.762·43-s + 1.34·45-s − 0.875·47-s + 2/7·49-s − 0.824·53-s + 0.390·59-s + 0.640·61-s + 1.13·63-s + 1.48·65-s + 0.488·67-s + 1.89·71-s + 0.468·73-s + 0.112·79-s + 81-s + 0.329·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107918963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107918963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−13.27333726592523, −12.98711850464624, −12.34743304077758, −12.14467329596964, −11.57202512388390, −11.05748709764806, −10.83672282651068, −9.964738064149203, −9.471050593977723, −9.260989386921226, −8.505819950162968, −8.021580862090105, −7.638694032050540, −6.927753240413011, −6.790714002050617, −6.002700340672127, −5.331732316042476, −4.967076646386116, −4.306962033540461, −3.567721201077844, −3.193363994547375, −2.831389727216260, −2.079113464781044, −0.8123638389302318, −0.4370423235832342,
0.4370423235832342, 0.8123638389302318, 2.079113464781044, 2.831389727216260, 3.193363994547375, 3.567721201077844, 4.306962033540461, 4.967076646386116, 5.331732316042476, 6.002700340672127, 6.790714002050617, 6.927753240413011, 7.638694032050540, 8.021580862090105, 8.505819950162968, 9.260989386921226, 9.471050593977723, 9.964738064149203, 10.83672282651068, 11.05748709764806, 11.57202512388390, 12.14467329596964, 12.34743304077758, 12.98711850464624, 13.27333726592523
