| L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s + 3·13-s − 17-s + 19-s − 21-s − 7·23-s − 27-s + 6·29-s − 7·31-s − 4·33-s − 10·37-s − 3·39-s − 5·41-s − 6·43-s − 6·49-s + 51-s + 2·53-s − 57-s − 8·59-s − 8·61-s + 63-s + 4·67-s + 7·69-s − 7·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.832·13-s − 0.242·17-s + 0.229·19-s − 0.218·21-s − 1.45·23-s − 0.192·27-s + 1.11·29-s − 1.25·31-s − 0.696·33-s − 1.64·37-s − 0.480·39-s − 0.780·41-s − 0.914·43-s − 6/7·49-s + 0.140·51-s + 0.274·53-s − 0.132·57-s − 1.04·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s + 0.842·69-s − 0.830·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.386343425\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.386343425\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 7 T + p T^{2} \) | 1.71.h |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−12.29892958524349, −11.88471390635988, −11.71574686178853, −11.12062894082457, −10.73257232698236, −10.24045211238674, −9.864173326150024, −9.238412678227526, −8.846820384071817, −8.386344956115996, −7.973074816310021, −7.332345354026832, −6.810623819043242, −6.486540225913720, −5.986569590736012, −5.585576411010376, −4.898063193795323, −4.570733693365864, −3.899344096458957, −3.551781604228973, −3.040865642508639, −1.965611926152306, −1.703837972335379, −1.192419607468025, −0.3199216562672963,
0.3199216562672963, 1.192419607468025, 1.703837972335379, 1.965611926152306, 3.040865642508639, 3.551781604228973, 3.899344096458957, 4.570733693365864, 4.898063193795323, 5.585576411010376, 5.986569590736012, 6.486540225913720, 6.810623819043242, 7.332345354026832, 7.973074816310021, 8.386344956115996, 8.846820384071817, 9.238412678227526, 9.864173326150024, 10.24045211238674, 10.73257232698236, 11.12062894082457, 11.71574686178853, 11.88471390635988, 12.29892958524349
