| L(s) = 1 | + 7-s − 3·9-s + 4·13-s − 4·17-s + 4·19-s + 8·23-s − 2·29-s + 8·31-s − 8·37-s + 6·41-s − 8·43-s + 8·47-s + 49-s − 4·59-s + 6·61-s − 3·63-s − 8·67-s − 12·71-s + 4·73-s + 4·79-s + 9·81-s − 10·89-s + 4·91-s + 12·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.10·13-s − 0.970·17-s + 0.917·19-s + 1.66·23-s − 0.371·29-s + 1.43·31-s − 1.31·37-s + 0.937·41-s − 1.21·43-s + 1.16·47-s + 1/7·49-s − 0.520·59-s + 0.768·61-s − 0.377·63-s − 0.977·67-s − 1.42·71-s + 0.468·73-s + 0.450·79-s + 81-s − 1.05·89-s + 0.419·91-s + 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.131825039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.131825039\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−16.48173262907472, −15.83006279936885, −15.36435543702186, −14.82477484660756, −14.10005291794583, −13.61317185327013, −13.26030172540948, −12.41662891229949, −11.65175610885863, −11.37324541751932, −10.75855067716959, −10.22149900841071, −9.125651715115609, −8.918062777702314, −8.315901470847413, −7.586579007863097, −6.839399086099265, −6.241165434786488, −5.493998899492572, −4.951679084643686, −4.126999160595671, −3.240420308981614, −2.704642045123169, −1.614871636501960, −0.6975934549091469,
0.6975934549091469, 1.614871636501960, 2.704642045123169, 3.240420308981614, 4.126999160595671, 4.951679084643686, 5.493998899492572, 6.241165434786488, 6.839399086099265, 7.586579007863097, 8.315901470847413, 8.918062777702314, 9.125651715115609, 10.22149900841071, 10.75855067716959, 11.37324541751932, 11.65175610885863, 12.41662891229949, 13.26030172540948, 13.61317185327013, 14.10005291794583, 14.82477484660756, 15.36435543702186, 15.83006279936885, 16.48173262907472
