| L(s) = 1 | − 2-s − 4-s + 3·8-s − 3·9-s − 11-s + 2·13-s − 16-s + 6·17-s + 3·18-s + 4·19-s + 22-s − 4·23-s − 2·26-s + 6·29-s + 8·31-s − 5·32-s − 6·34-s + 3·36-s + 2·37-s − 4·38-s − 2·41-s − 4·43-s + 44-s + 4·46-s − 12·47-s − 2·52-s + 2·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 9-s − 0.301·11-s + 0.554·13-s − 1/4·16-s + 1.45·17-s + 0.707·18-s + 0.917·19-s + 0.213·22-s − 0.834·23-s − 0.392·26-s + 1.11·29-s + 1.43·31-s − 0.883·32-s − 1.02·34-s + 1/2·36-s + 0.328·37-s − 0.648·38-s − 0.312·41-s − 0.609·43-s + 0.150·44-s + 0.589·46-s − 1.75·47-s − 0.277·52-s + 0.274·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.163793263\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.163793263\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−16.21361266212595, −15.94482520821382, −15.07971252524364, −14.34251297493581, −13.93981114650197, −13.64556844224972, −12.80130364991736, −12.17215974342080, −11.58334156021147, −11.06982926561717, −10.14621685764985, −9.959421815840638, −9.359079389996512, −8.486362289797972, −8.071642184042931, −7.893383176233337, −6.765933021861523, −6.171893252246432, −5.210359031551162, −5.063201362031803, −3.886733232403702, −3.317411017030572, −2.467745551612941, −1.302377808421364, −0.6104646609088298,
0.6104646609088298, 1.302377808421364, 2.467745551612941, 3.317411017030572, 3.886733232403702, 5.063201362031803, 5.210359031551162, 6.171893252246432, 6.765933021861523, 7.893383176233337, 8.071642184042931, 8.486362289797972, 9.359079389996512, 9.959421815840638, 10.14621685764985, 11.06982926561717, 11.58334156021147, 12.17215974342080, 12.80130364991736, 13.64556844224972, 13.93981114650197, 14.34251297493581, 15.07971252524364, 15.94482520821382, 16.21361266212595
