| L(s) = 1 | − 3-s − 7-s + 9-s + 11-s − 6·13-s + 4·17-s − 6·19-s + 21-s − 4·23-s − 5·25-s − 27-s − 6·29-s − 2·31-s − 33-s − 10·37-s + 6·39-s − 4·41-s − 8·43-s − 6·47-s + 49-s − 4·51-s + 10·53-s + 6·57-s + 2·61-s − 63-s + 4·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s + 0.970·17-s − 1.37·19-s + 0.218·21-s − 0.834·23-s − 25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s − 0.174·33-s − 1.64·37-s + 0.960·39-s − 0.624·41-s − 1.21·43-s − 0.875·47-s + 1/7·49-s − 0.560·51-s + 1.37·53-s + 0.794·57-s + 0.256·61-s − 0.125·63-s + 0.488·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4566950393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4566950393\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.20799473839888, −15.43469689394395, −15.03126825889454, −14.46034134519295, −13.90414079854072, −13.14228736080014, −12.62074551028505, −12.06401886257965, −11.79146050138636, −10.96888528137834, −10.29897569790357, −9.811285036008934, −9.498546477156712, −8.459840587176474, −8.005011225436344, −7.061629736916618, −6.890046108398232, −5.950457595297236, −5.420439295479824, −4.821479151603292, −3.939799815321395, −3.438578277210884, −2.253023657286273, −1.752309568929667, −0.2894266947289624,
0.2894266947289624, 1.752309568929667, 2.253023657286273, 3.438578277210884, 3.939799815321395, 4.821479151603292, 5.420439295479824, 5.950457595297236, 6.890046108398232, 7.061629736916618, 8.005011225436344, 8.459840587176474, 9.498546477156712, 9.811285036008934, 10.29897569790357, 10.96888528137834, 11.79146050138636, 12.06401886257965, 12.62074551028505, 13.14228736080014, 13.90414079854072, 14.46034134519295, 15.03126825889454, 15.43469689394395, 16.20799473839888
