| L(s) = 1 | − 3-s − 3·5-s + 7-s + 9-s − 3·13-s + 3·15-s + 4·17-s − 19-s − 21-s − 4·23-s + 4·25-s − 27-s − 3·29-s + 2·31-s − 3·35-s − 3·37-s + 3·39-s − 4·43-s − 3·45-s − 3·47-s + 49-s − 4·51-s − 10·53-s + 57-s − 3·59-s − 2·61-s + 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.832·13-s + 0.774·15-s + 0.970·17-s − 0.229·19-s − 0.218·21-s − 0.834·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s + 0.359·31-s − 0.507·35-s − 0.493·37-s + 0.480·39-s − 0.609·43-s − 0.447·45-s − 0.437·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s + 0.132·57-s − 0.390·59-s − 0.256·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4671488565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4671488565\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.85356485322165, −14.30628801856317, −13.86729413765166, −13.01788382999608, −12.45554216440131, −12.12484638156728, −11.71174627287474, −11.17804026237002, −10.75656071995368, −9.938322086196672, −9.746275101569355, −8.790047798811418, −8.223985909278475, −7.712387376765867, −7.407425395966660, −6.719216937321627, −6.044619138504957, −5.384896889994812, −4.769618326365851, −4.327060136097838, −3.616959201921324, −3.088656793858744, −2.085859839772796, −1.294824724484759, −0.2715930316868101,
0.2715930316868101, 1.294824724484759, 2.085859839772796, 3.088656793858744, 3.616959201921324, 4.327060136097838, 4.769618326365851, 5.384896889994812, 6.044619138504957, 6.719216937321627, 7.407425395966660, 7.712387376765867, 8.223985909278475, 8.790047798811418, 9.746275101569355, 9.938322086196672, 10.75656071995368, 11.17804026237002, 11.71174627287474, 12.12484638156728, 12.45554216440131, 13.01788382999608, 13.86729413765166, 14.30628801856317, 14.85356485322165
