| L(s) = 1 | + (0.415 − 0.909i)2-s + (−0.959 − 0.281i)3-s + (−0.654 − 0.755i)4-s + (−0.627 + 0.403i)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 + (0.106 + 0.738i)10-s + (−0.872 − 1.91i)11-s + (0.415 + 0.909i)12-s + (−0.471 − 3.27i)13-s + (0.841 + 0.540i)14-s + (0.715 − 0.210i)15-s + (−0.142 + 0.989i)16-s + (−1.19 + 1.37i)17-s + ⋯ |
| L(s) = 1 | + (0.293 − 0.643i)2-s + (−0.553 − 0.162i)3-s + (−0.327 − 0.377i)4-s + (−0.280 + 0.180i)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 + (0.0335 + 0.233i)10-s + (−0.263 − 0.576i)11-s + (0.119 + 0.262i)12-s + (−0.130 − 0.908i)13-s + (0.224 + 0.144i)14-s + (0.184 − 0.0542i)15-s + (−0.0355 + 0.247i)16-s + (−0.289 + 0.334i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0912024 + 0.128791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0912024 + 0.128791i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.133 - 4.79i)T \) |
| good | 5 | \( 1 + (0.627 - 0.403i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (0.872 + 1.91i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.471 + 3.27i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (1.19 - 1.37i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.06 + 2.38i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (5.49 - 6.34i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (6.69 - 1.96i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (0.528 + 0.339i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (4.26 - 2.73i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (4.63 + 1.35i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 - 8.15T + 47T^{2} \) |
| 53 | \( 1 + (1.14 - 7.93i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.895 + 6.23i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-4.25 + 1.24i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 2.87i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (2.20 - 4.83i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (2.28 + 2.64i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (-1.05 - 7.34i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (6.36 + 4.08i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-9.52 - 2.79i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (10.4 - 6.72i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.65888362774479476980050391442, −9.544405131936639433591573697613, −8.750010037938946434981551746873, −7.72441532277260169033400431228, −6.82408105656485733468628190068, −5.61896734023791743973235357696, −5.25564405437375529779066473441, −3.86052583913499623625923339234, −3.00424918883991056424546131569, −1.63048842033674416348525768164,
0.06902353447038005929623022913, 2.14945281193385171631107198300, 3.91857790000686783847772323238, 4.42079195963063375413957515711, 5.42283371021289199876443947251, 6.38768000966093679034152550598, 7.10108277728483360248529066327, 7.912525267837068326800201204583, 8.876993841682243401454417048950, 9.757686482876032675094919632539
