| L(s) = 1 | + (−0.104 + 0.994i)2-s + (1.65 + 0.509i)3-s + (−0.978 − 0.207i)4-s + (−0.0756 + 0.169i)5-s + (−0.679 + 1.59i)6-s + (1.63 + 2.07i)7-s + (0.309 − 0.951i)8-s + (2.48 + 1.68i)9-s + (−0.161 − 0.0930i)10-s + (−1.90 + 2.71i)11-s + (−1.51 − 0.842i)12-s + (0.557 − 0.766i)13-s + (−2.23 + 1.41i)14-s + (−0.211 + 0.242i)15-s + (0.913 + 0.406i)16-s + (−0.846 − 8.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.955 + 0.294i)3-s + (−0.489 − 0.103i)4-s + (−0.0338 + 0.0760i)5-s + (−0.277 + 0.650i)6-s + (0.618 + 0.785i)7-s + (0.109 − 0.336i)8-s + (0.827 + 0.562i)9-s + (−0.0509 − 0.0294i)10-s + (−0.574 + 0.818i)11-s + (−0.436 − 0.243i)12-s + (0.154 − 0.212i)13-s + (−0.598 + 0.377i)14-s + (−0.0546 + 0.0626i)15-s + (0.228 + 0.101i)16-s + (−0.205 − 1.95i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.146 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18569 + 1.37482i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18569 + 1.37482i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-1.65 - 0.509i)T \) |
| 7 | \( 1 + (-1.63 - 2.07i)T \) |
| 11 | \( 1 + (1.90 - 2.71i)T \) |
| good | 5 | \( 1 + (0.0756 - 0.169i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-0.557 + 0.766i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.846 + 8.05i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 4.81i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (0.938 - 0.542i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.413 - 1.27i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.24 - 1.89i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-2.32 + 2.58i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.18 - 3.66i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 11.0iT - 43T^{2} \) |
| 47 | \( 1 + (0.589 + 2.77i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (2.35 + 5.29i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.705 + 3.32i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-5.92 + 13.3i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-0.0659 + 0.114i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.19 - 7.15i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.96 + 13.9i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (12.4 + 1.30i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (5.65 - 4.11i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.52 + 0.881i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.39 - 5.37i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.20175080514036226520423827983, −10.04690045114325044220449433328, −9.342974285426538025941582221707, −8.541555549348140363324432376625, −7.67525735656903555380481100641, −7.02652781046965582222476897364, −5.39249457922511946326280163709, −4.80178422928366338868938298920, −3.34977671626011166540927977280, −2.04357925918499976535098488999,
1.19056247372906691512692424553, 2.53721504704105002879768724820, 3.74544432662336186976110764907, 4.59620235058904390118410004471, 6.19305545471339933893467028127, 7.45274122203223484773923416667, 8.309839406284963743984171548208, 8.823595379329491037750331053897, 10.05607946061566264712190686260, 10.75626419223252404525401107393
