| L(s) = 1 | + (0.415 + 0.909i)2-s + (−0.959 + 0.281i)3-s + (−0.654 + 0.755i)4-s + (−0.114 − 0.0736i)5-s + (−0.654 − 0.755i)6-s + (0.142 + 0.989i)7-s + (−0.959 − 0.281i)8-s + (0.841 − 0.540i)9-s + (0.0193 − 0.134i)10-s + (−0.794 + 1.73i)11-s + (0.415 − 0.909i)12-s + (−0.231 + 1.61i)13-s + (−0.841 + 0.540i)14-s + (0.130 + 0.0384i)15-s + (−0.142 − 0.989i)16-s + (−2.09 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.293 + 0.643i)2-s + (−0.553 + 0.162i)3-s + (−0.327 + 0.377i)4-s + (−0.0512 − 0.0329i)5-s + (−0.267 − 0.308i)6-s + (0.0537 + 0.374i)7-s + (−0.339 − 0.0996i)8-s + (0.280 − 0.180i)9-s + (0.00613 − 0.0426i)10-s + (−0.239 + 0.524i)11-s + (0.119 − 0.262i)12-s + (−0.0642 + 0.446i)13-s + (−0.224 + 0.144i)14-s + (0.0337 + 0.00991i)15-s + (−0.0355 − 0.247i)16-s + (−0.508 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.427i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.121995 - 0.543404i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121995 - 0.543404i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.415 - 0.909i)T \) |
| 3 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (-0.142 - 0.989i)T \) |
| 23 | \( 1 + (3.50 - 3.27i)T \) |
| good | 5 | \( 1 + (0.114 + 0.0736i)T + (2.07 + 4.54i)T^{2} \) |
| 11 | \( 1 + (0.794 - 1.73i)T + (-7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (0.231 - 1.61i)T + (-12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (2.09 + 2.41i)T + (-2.41 + 16.8i)T^{2} \) |
| 19 | \( 1 + (2.38 - 2.75i)T + (-2.70 - 18.8i)T^{2} \) |
| 29 | \( 1 + (-2.27 - 2.62i)T + (-4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-4.47 - 1.31i)T + (26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (5.69 - 3.66i)T + (15.3 - 33.6i)T^{2} \) |
| 41 | \( 1 + (9.25 + 5.95i)T + (17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (2.86 - 0.842i)T + (36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + (-0.287 - 2.00i)T + (-50.8 + 14.9i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 14.2i)T + (-56.6 - 16.6i)T^{2} \) |
| 61 | \( 1 + (7.00 + 2.05i)T + (51.3 + 32.9i)T^{2} \) |
| 67 | \( 1 + (-3.63 - 7.95i)T + (-43.8 + 50.6i)T^{2} \) |
| 71 | \( 1 + (-1.77 - 3.89i)T + (-46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (-1.96 + 2.26i)T + (-10.3 - 72.2i)T^{2} \) |
| 79 | \( 1 + (0.304 - 2.11i)T + (-75.7 - 22.2i)T^{2} \) |
| 83 | \( 1 + (-5.07 + 3.25i)T + (34.4 - 75.4i)T^{2} \) |
| 89 | \( 1 + (9.13 - 2.68i)T + (74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-9.61 - 6.18i)T + (40.2 + 88.2i)T^{2} \) |
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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
−10.31663457027611789140638310110, −9.739598988299043080141202359036, −8.664694504948762828967119797598, −7.969114859942116513235055868407, −6.84787517071866800192290337205, −6.32908028519273732685177254182, −5.21675528001478621311538166615, −4.62224125907736924957278156301, −3.51476143924556708509659727662, −1.98877914289339548278825722124,
0.24095209845849736485811634215, 1.77475596654265905079162009393, 3.07730510892158786992582621205, 4.18760696889894243329631665504, 5.03506139861086401445947008475, 6.04269643270788850180203877677, 6.77969008986978703159979843605, 7.981653738108488419772317971780, 8.697518254321335709041950773492, 9.911570557569452459134358443209
