| L(s) = 1 | + (−0.959 + 0.281i)2-s + (1.19 + 1.37i)3-s + (0.841 − 0.540i)4-s + (0.248 + 1.73i)5-s + (−1.52 − 0.982i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.0432 + 0.300i)9-s + (−0.726 − 1.59i)10-s + (0.896 + 0.263i)11-s + (1.74 + 0.512i)12-s + (2.78 + 6.09i)13-s + (0.142 − 0.989i)14-s + (−2.08 + 2.40i)15-s + (0.415 − 0.909i)16-s + (−6.32 − 4.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (0.687 + 0.793i)3-s + (0.420 − 0.270i)4-s + (0.111 + 0.774i)5-s + (−0.624 − 0.401i)6-s + (−0.157 + 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.0144 + 0.100i)9-s + (−0.229 − 0.503i)10-s + (0.270 + 0.0793i)11-s + (0.503 + 0.147i)12-s + (0.771 + 1.68i)13-s + (0.0380 − 0.264i)14-s + (−0.537 + 0.620i)15-s + (0.103 − 0.227i)16-s + (−1.53 − 0.985i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734209 + 0.924322i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734209 + 0.924322i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.64 + 1.19i)T \) |
| good | 3 | \( 1 + (-1.19 - 1.37i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (-0.248 - 1.73i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.896 - 0.263i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.78 - 6.09i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (6.32 + 4.06i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.74 - 1.11i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (2.21 + 1.42i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (6.49 - 7.50i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.435 + 3.03i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.15 - 8.00i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-3.49 - 4.03i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 1.68T + 47T^{2} \) |
| 53 | \( 1 + (-5.55 + 12.1i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (3.18 + 6.97i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-4.71 + 5.43i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-9.43 + 2.77i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.07 + 2.07i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-10.8 + 6.94i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (0.418 + 0.916i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (1.27 - 8.84i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (11.4 + 13.1i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (1.61 + 11.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.33755967080857797768819355655, −10.99172672169760696887089055153, −9.691635382430924644240203015373, −9.131519809735793257403217838885, −8.543094193975477072836483946369, −6.86128493893030461897746258749, −6.54449865400818470166853017820, −4.70946005637435790967694491013, −3.47722921642058864170596283105, −2.19547716228021400069805984596,
1.04449043167929019259401788373, 2.42063148783579566623642166621, 3.87670329589017840395382654798, 5.54490259370748172919919549924, 6.82936848629088816921590567653, 7.76290757964701407005884990228, 8.631441389234655877932746640694, 9.076848537955813189146812516334, 10.57122663717611028522666475706, 11.05976930847401280977866494365
