| L(s) = 1 | + 2-s + 4-s + 2·5-s − 4·7-s + 8-s + 2·10-s + 11-s + 6·13-s − 4·14-s + 16-s + 4·19-s + 2·20-s + 22-s − 25-s + 6·26-s − 4·28-s + 2·29-s + 32-s − 8·35-s + 10·37-s + 4·38-s + 2·40-s − 10·41-s − 4·43-s + 44-s − 4·47-s + 9·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.51·7-s + 0.353·8-s + 0.632·10-s + 0.301·11-s + 1.66·13-s − 1.06·14-s + 1/4·16-s + 0.917·19-s + 0.447·20-s + 0.213·22-s − 1/5·25-s + 1.17·26-s − 0.755·28-s + 0.371·29-s + 0.176·32-s − 1.35·35-s + 1.64·37-s + 0.648·38-s + 0.316·40-s − 1.56·41-s − 0.609·43-s + 0.150·44-s − 0.583·47-s + 9/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57222 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.866520743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.866520743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.15094464438377, −13.68523765382361, −13.35676457300424, −13.11060073895297, −12.47717249840290, −11.89879410911314, −11.37854666373453, −10.85449507959580, −10.16595762261972, −9.757301860306611, −9.413874771586172, −8.714921412785093, −8.154657596605943, −7.404074082608136, −6.593028484135767, −6.422402012430473, −5.979265042929856, −5.406591717267107, −4.770692554104388, −3.797363924970931, −3.546162688871516, −2.960375901991811, −2.212260364536403, −1.430486332743373, −0.6925634293655071,
0.6925634293655071, 1.430486332743373, 2.212260364536403, 2.960375901991811, 3.546162688871516, 3.797363924970931, 4.770692554104388, 5.406591717267107, 5.979265042929856, 6.422402012430473, 6.593028484135767, 7.404074082608136, 8.154657596605943, 8.714921412785093, 9.413874771586172, 9.757301860306611, 10.16595762261972, 10.85449507959580, 11.37854666373453, 11.89879410911314, 12.47717249840290, 13.11060073895297, 13.35676457300424, 13.68523765382361, 14.15094464438377
