| L(s) = 1 | − 2-s − 4-s + 5-s − 2·7-s + 3·8-s − 10-s + 4·11-s − 13-s + 2·14-s − 16-s + 6·19-s − 20-s − 4·22-s + 25-s + 26-s + 2·28-s + 4·29-s + 10·31-s − 5·32-s − 2·35-s + 2·37-s − 6·38-s + 3·40-s + 6·41-s − 8·43-s − 4·44-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 0.755·7-s + 1.06·8-s − 0.316·10-s + 1.20·11-s − 0.277·13-s + 0.534·14-s − 1/4·16-s + 1.37·19-s − 0.223·20-s − 0.852·22-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 0.742·29-s + 1.79·31-s − 0.883·32-s − 0.338·35-s + 0.328·37-s − 0.973·38-s + 0.474·40-s + 0.937·41-s − 1.21·43-s − 0.603·44-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.707879410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.707879410\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−13.15562907943187, −12.97197072330562, −12.15569315274595, −11.76991090484766, −11.43936415342995, −10.63512940267122, −10.09235388020171, −9.817351196137411, −9.512592982309749, −9.001844093955382, −8.559948342098576, −7.980515327301538, −7.548407522238545, −6.800469549121792, −6.551598611548253, −5.983807337645980, −5.315063571386734, −4.708952718682890, −4.379624076038877, −3.544930181337479, −3.166132994368020, −2.460393669785403, −1.568396513754148, −1.102065587485371, −0.5032882199102366,
0.5032882199102366, 1.102065587485371, 1.568396513754148, 2.460393669785403, 3.166132994368020, 3.544930181337479, 4.379624076038877, 4.708952718682890, 5.315063571386734, 5.983807337645980, 6.551598611548253, 6.800469549121792, 7.548407522238545, 7.980515327301538, 8.559948342098576, 9.001844093955382, 9.512592982309749, 9.817351196137411, 10.09235388020171, 10.63512940267122, 11.43936415342995, 11.76991090484766, 12.15569315274595, 12.97197072330562, 13.15562907943187
