| L(s) = 1 | + (−0.340 − 1.40i)2-s + (0.327 + 0.945i)3-s + (−0.0785 + 0.0405i)4-s + (−1.83 + 0.735i)5-s + (1.21 − 0.781i)6-s + (−2.55 − 0.683i)7-s + (−1.80 − 2.08i)8-s + (−0.786 + 0.618i)9-s + (1.65 + 2.32i)10-s + (−1.43 + 5.90i)11-s + (−0.0639 − 0.0610i)12-s + (0.301 − 0.659i)13-s + (−0.0885 + 3.82i)14-s + (−1.29 − 1.49i)15-s + (−2.41 + 3.39i)16-s + (0.167 + 3.51i)17-s + ⋯ |
| L(s) = 1 | + (−0.240 − 0.993i)2-s + (0.188 + 0.545i)3-s + (−0.0392 + 0.0202i)4-s + (−0.821 + 0.328i)5-s + (0.496 − 0.318i)6-s + (−0.966 − 0.258i)7-s + (−0.639 − 0.738i)8-s + (−0.262 + 0.206i)9-s + (0.524 + 0.736i)10-s + (−0.432 + 1.78i)11-s + (−0.0184 − 0.0176i)12-s + (0.0835 − 0.183i)13-s + (−0.0236 + 1.02i)14-s + (−0.334 − 0.386i)15-s + (−0.604 + 0.848i)16-s + (0.0405 + 0.851i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.143 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.247650 + 0.286187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.247650 + 0.286187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.327 - 0.945i)T \) |
| 7 | \( 1 + (2.55 + 0.683i)T \) |
| 23 | \( 1 + (-2.06 - 4.32i)T \) |
| good | 2 | \( 1 + (0.340 + 1.40i)T + (-1.77 + 0.916i)T^{2} \) |
| 5 | \( 1 + (1.83 - 0.735i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (1.43 - 5.90i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (-0.301 + 0.659i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.167 - 3.51i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.237 + 4.98i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (6.50 - 4.18i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (8.43 - 1.62i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (4.28 - 3.36i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.950 + 6.61i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.28 + 6.10i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (0.891 + 1.54i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-9.57 - 0.914i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (4.81 + 6.75i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (1.97 - 5.71i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (-5.05 + 4.81i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (11.9 + 3.51i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-7.17 + 3.69i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (-8.01 + 0.765i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (-1.36 - 9.51i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (3.33 + 0.642i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (-0.274 + 1.91i)T + (-93.0 - 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.88146975505518967481110195513, −10.56939613299142445708264178538, −9.565455210565586098635081195592, −9.061195626863428208247833644379, −7.40209945840447420299789399574, −6.97244557881233207344133103981, −5.40427901797797474218876204560, −3.95414892722569135067431194950, −3.34295177049415111602164772498, −2.04953675841202712149760208463,
0.22339603305820781650062308693, 2.71408936817632444266702790789, 3.73073822981202925848473435596, 5.59613342037992138465482359330, 6.12535692599926755210816783209, 7.21222385188502588345238019294, 7.941188886580994031551036012436, 8.629037204375198134454230917847, 9.402473951087739757208619687794, 10.94075422493296224033819720423
