| L(s) = 1 | − 3-s − 5-s + 9-s − 4·11-s + 6·13-s + 15-s − 2·17-s + 4·19-s + 25-s − 27-s − 10·29-s + 4·31-s + 4·33-s + 10·37-s − 6·39-s − 2·41-s + 4·43-s − 45-s − 8·47-s + 2·51-s − 2·53-s + 4·55-s − 4·57-s + 12·59-s − 10·61-s − 6·65-s − 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 0.258·15-s − 0.485·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.696·33-s + 1.64·37-s − 0.960·39-s − 0.312·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 0.280·51-s − 0.274·53-s + 0.539·55-s − 0.529·57-s + 1.56·59-s − 1.28·61-s − 0.744·65-s − 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.317679251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.317679251\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−14.85035345369582, −13.89559203595918, −13.43254962444285, −13.03489415891087, −12.72895563323308, −11.80065332457790, −11.36989352894638, −11.18988714853422, −10.46019720633538, −10.11470820206391, −9.263611378036390, −8.904390417404718, −8.055650967203967, −7.790438847308128, −7.201685221835274, −6.468836049757668, −5.866072952128844, −5.561010684231077, −4.725818502420768, −4.269377327665266, −3.480229634012592, −3.004164266545327, −2.064169863533602, −1.252042283606037, −0.4569763055114531,
0.4569763055114531, 1.252042283606037, 2.064169863533602, 3.004164266545327, 3.480229634012592, 4.269377327665266, 4.725818502420768, 5.561010684231077, 5.866072952128844, 6.468836049757668, 7.201685221835274, 7.790438847308128, 8.055650967203967, 8.904390417404718, 9.263611378036390, 10.11470820206391, 10.46019720633538, 11.18988714853422, 11.36989352894638, 11.80065332457790, 12.72895563323308, 13.03489415891087, 13.43254962444285, 13.89559203595918, 14.85035345369582
