| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 2·11-s − 4·13-s − 2·14-s + 16-s + 2·17-s − 20-s − 2·22-s − 4·23-s + 25-s − 4·26-s − 2·28-s + 8·31-s + 32-s + 2·34-s + 2·35-s − 8·37-s − 40-s − 8·41-s − 6·43-s − 2·44-s − 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.603·11-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.485·17-s − 0.223·20-s − 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.377·28-s + 1.43·31-s + 0.176·32-s + 0.342·34-s + 0.338·35-s − 1.31·37-s − 0.158·40-s − 1.24·41-s − 0.914·43-s − 0.301·44-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.483492797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483492797\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−15.14991731757185, −14.44518416515092, −13.97505732228994, −13.44902921459128, −12.96877808099376, −12.21631812921665, −12.09164100082621, −11.62979309644650, −10.59299661276100, −10.39394564350825, −9.784599642996881, −9.241838461610411, −8.248665760588670, −8.031243339611061, −7.186421090035987, −6.842585332967279, −6.161235056206649, −5.484325880935126, −4.942037861531447, −4.368328078425023, −3.597683832872735, −3.058109216743145, −2.477670282305718, −1.620535079606976, −0.3907016936962984,
0.3907016936962984, 1.620535079606976, 2.477670282305718, 3.058109216743145, 3.597683832872735, 4.368328078425023, 4.942037861531447, 5.484325880935126, 6.161235056206649, 6.842585332967279, 7.186421090035987, 8.031243339611061, 8.248665760588670, 9.241838461610411, 9.784599642996881, 10.39394564350825, 10.59299661276100, 11.62979309644650, 12.09164100082621, 12.21631812921665, 12.96877808099376, 13.44902921459128, 13.97505732228994, 14.44518416515092, 15.14991731757185
