| L(s) = 1 | + 3-s − 2·5-s + 7-s + 9-s + 4·11-s + 4·13-s − 2·15-s − 4·17-s + 2·19-s + 21-s + 23-s − 25-s + 27-s + 6·29-s + 6·31-s + 4·33-s − 2·35-s + 2·37-s + 4·39-s − 10·41-s − 8·43-s − 2·45-s + 10·47-s + 49-s − 4·51-s + 6·53-s − 8·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.10·13-s − 0.516·15-s − 0.970·17-s + 0.458·19-s + 0.218·21-s + 0.208·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s − 0.338·35-s + 0.328·37-s + 0.640·39-s − 1.56·41-s − 1.21·43-s − 0.298·45-s + 1.45·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.204447139\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.204447139\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−15.04950177382357, −14.72522704604903, −13.95750433774378, −13.52748603804555, −13.27108581707908, −12.20564027204037, −11.91987982654573, −11.44783391543252, −10.98415720069463, −10.16963337620662, −9.770126205782553, −8.807764345405357, −8.579276386588592, −8.276907210543661, −7.381910963882535, −6.889660938489164, −6.410481440758675, −5.642094582827144, −4.720717652961308, −4.228127919337206, −3.726547991796055, −3.139769291624553, −2.254902487510010, −1.401141393391046, −0.7198206832784289,
0.7198206832784289, 1.401141393391046, 2.254902487510010, 3.139769291624553, 3.726547991796055, 4.228127919337206, 4.720717652961308, 5.642094582827144, 6.410481440758675, 6.889660938489164, 7.381910963882535, 8.276907210543661, 8.579276386588592, 8.807764345405357, 9.770126205782553, 10.16963337620662, 10.98415720069463, 11.44783391543252, 11.91987982654573, 12.20564027204037, 13.27108581707908, 13.52748603804555, 13.95750433774378, 14.72522704604903, 15.04950177382357
