| L(s) = 1 | − 2·2-s + 2·4-s − 5-s + 2·7-s + 2·10-s + 3·11-s + 13-s − 4·14-s − 4·16-s + 7·19-s − 2·20-s − 6·22-s − 4·23-s + 25-s − 2·26-s + 4·28-s + 29-s − 4·31-s + 8·32-s − 2·35-s + 10·37-s − 14·38-s + 7·41-s − 4·43-s + 6·44-s + 8·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s + 0.755·7-s + 0.632·10-s + 0.904·11-s + 0.277·13-s − 1.06·14-s − 16-s + 1.60·19-s − 0.447·20-s − 1.27·22-s − 0.834·23-s + 1/5·25-s − 0.392·26-s + 0.755·28-s + 0.185·29-s − 0.718·31-s + 1.41·32-s − 0.338·35-s + 1.64·37-s − 2.27·38-s + 1.09·41-s − 0.609·43-s + 0.904·44-s + 1.17·46-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.63335410337222, −12.91504247032322, −12.26993820876474, −11.82304277622489, −11.43972186213247, −11.03571031612680, −10.67933747305480, −9.936194418309777, −9.531407141433049, −9.257508146739846, −8.648584755552768, −8.209951179810545, −7.715329616891715, −7.434975612493064, −6.955323013426861, −6.179051113049193, −5.815873470971920, −5.017001527098035, −4.407764867601170, −4.072715395352682, −3.274598329618160, −2.659682681134051, −1.828340978630944, −1.319040622653269, −0.8861031022081992, 0,
0.8861031022081992, 1.319040622653269, 1.828340978630944, 2.659682681134051, 3.274598329618160, 4.072715395352682, 4.407764867601170, 5.017001527098035, 5.815873470971920, 6.179051113049193, 6.955323013426861, 7.434975612493064, 7.715329616891715, 8.209951179810545, 8.648584755552768, 9.257508146739846, 9.531407141433049, 9.936194418309777, 10.67933747305480, 11.03571031612680, 11.43972186213247, 11.82304277622489, 12.26993820876474, 12.91504247032322, 13.63335410337222
