| L(s) = 1 | − 5-s + 7-s + 11-s + 6·13-s − 6·17-s + 4·23-s + 25-s + 2·29-s − 35-s + 2·37-s + 2·41-s + 8·43-s − 12·47-s + 49-s − 6·53-s − 55-s + 4·59-s − 2·61-s − 6·65-s + 12·67-s + 4·71-s − 6·73-s + 77-s + 8·79-s + 12·83-s + 6·85-s + 6·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.301·11-s + 1.66·13-s − 1.45·17-s + 0.834·23-s + 1/5·25-s + 0.371·29-s − 0.169·35-s + 0.328·37-s + 0.312·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s − 0.134·55-s + 0.520·59-s − 0.256·61-s − 0.744·65-s + 1.46·67-s + 0.474·71-s − 0.702·73-s + 0.113·77-s + 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.969169472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.969169472\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.93469739037715, −12.65215749176044, −11.89899738958478, −11.46511932728950, −11.05364019008972, −10.88300277528659, −10.33172695358747, −9.489905450754679, −9.190620643557203, −8.702870275218066, −8.235006786013948, −7.926184328144383, −7.198900231151656, −6.675740016921603, −6.338807579341831, −5.855399497534444, −5.057518571889325, −4.690301465971471, −4.107919540137505, −3.651422484144638, −3.121910966498001, −2.390308802944421, −1.781117611438332, −1.091307086039004, −0.5390005026583258,
0.5390005026583258, 1.091307086039004, 1.781117611438332, 2.390308802944421, 3.121910966498001, 3.651422484144638, 4.107919540137505, 4.690301465971471, 5.057518571889325, 5.855399497534444, 6.338807579341831, 6.675740016921603, 7.198900231151656, 7.926184328144383, 8.235006786013948, 8.702870275218066, 9.190620643557203, 9.489905450754679, 10.33172695358747, 10.88300277528659, 11.05364019008972, 11.46511932728950, 11.89899738958478, 12.65215749176044, 12.93469739037715
