| L(s) = 1 | − 5-s − 2·7-s + 6·13-s − 7·17-s − 7·19-s − 7·23-s + 25-s + 6·29-s + 3·31-s + 2·35-s + 6·37-s − 4·41-s − 8·43-s + 4·47-s − 3·49-s − 5·53-s + 6·59-s + 3·61-s − 6·65-s + 10·67-s − 12·71-s + 16·73-s + 79-s + 9·83-s + 7·85-s + 4·89-s − 12·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 1.66·13-s − 1.69·17-s − 1.60·19-s − 1.45·23-s + 1/5·25-s + 1.11·29-s + 0.538·31-s + 0.338·35-s + 0.986·37-s − 0.624·41-s − 1.21·43-s + 0.583·47-s − 3/7·49-s − 0.686·53-s + 0.781·59-s + 0.384·61-s − 0.744·65-s + 1.22·67-s − 1.42·71-s + 1.87·73-s + 0.112·79-s + 0.987·83-s + 0.759·85-s + 0.423·89-s − 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.124643732\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.124643732\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.052643436285463272092774811965, −6.71849101624627224342716208120, −6.50914074431724494672086911819, −6.01215624018625476805505034043, −4.81501996717403061806759556205, −4.11471034282584466226709026220, −3.66383082169737615189817423210, −2.64141959696078601993663671360, −1.81783512963193332695468453073, −0.50122377764999178905470505185,
0.50122377764999178905470505185, 1.81783512963193332695468453073, 2.64141959696078601993663671360, 3.66383082169737615189817423210, 4.11471034282584466226709026220, 4.81501996717403061806759556205, 6.01215624018625476805505034043, 6.50914074431724494672086911819, 6.71849101624627224342716208120, 8.052643436285463272092774811965
