| L(s) = 1 | + (−1.19 − 0.887i)2-s + (−0.856 + 1.50i)3-s + (0.0595 + 0.199i)4-s + (3.59 + 2.36i)5-s + (2.35 − 1.03i)6-s + (0.605 − 2.57i)7-s + (−0.910 + 2.50i)8-s + (−1.53 − 2.57i)9-s + (−2.18 − 6.00i)10-s + (2.45 − 4.88i)11-s + (−0.350 − 0.0807i)12-s + (−0.235 − 0.101i)13-s + (−3.00 + 2.53i)14-s + (−6.63 + 3.38i)15-s + (3.65 − 2.40i)16-s + (0.337 + 0.283i)17-s + ⋯ |
| L(s) = 1 | + (−0.842 − 0.627i)2-s + (−0.494 + 0.869i)3-s + (0.0297 + 0.0995i)4-s + (1.60 + 1.05i)5-s + (0.962 − 0.422i)6-s + (0.228 − 0.973i)7-s + (−0.322 + 0.884i)8-s + (−0.511 − 0.859i)9-s + (−0.691 − 1.89i)10-s + (0.740 − 1.47i)11-s + (−0.101 − 0.0233i)12-s + (−0.0654 − 0.0282i)13-s + (−0.803 + 0.676i)14-s + (−1.71 + 0.874i)15-s + (0.913 − 0.600i)16-s + (0.0818 + 0.0686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.992019 - 0.303194i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.992019 - 0.303194i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.856 - 1.50i)T \) |
| 7 | \( 1 + (-0.605 + 2.57i)T \) |
| good | 2 | \( 1 + (1.19 + 0.887i)T + (0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (-3.59 - 2.36i)T + (1.98 + 4.59i)T^{2} \) |
| 11 | \( 1 + (-2.45 + 4.88i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (0.235 + 0.101i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (-0.337 - 0.283i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (1.78 + 2.12i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-1.40 - 1.32i)T + (1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-0.952 + 8.14i)T + (-28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (0.260 - 1.09i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (1.43 - 8.13i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-0.437 - 0.587i)T + (-11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (0.563 + 9.67i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (-11.0 + 2.62i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (-1.45 - 0.841i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.65 + 0.829i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-4.15 - 1.24i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (-7.26 + 0.849i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-2.79 - 7.68i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (1.13 - 3.11i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (5.64 - 7.58i)T + (-22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-1.31 + 1.76i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-8.87 - 3.23i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.56 + 13.0i)T + (-38.4 + 89.0i)T^{2} \) |
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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
−10.49413154914046441310767589001, −10.03710252666068069722970513070, −9.289345847251544598009789580443, −8.495725871111493468527176395375, −6.83212386551515104937738351514, −6.04565305007856484856005011246, −5.28539618588893118723711569621, −3.70631855764133150120008640511, −2.54904830007351983822440138615, −0.988376287067134150225945706256,
1.33278447869964500176473274544, 2.25439639036409065464878182099, 4.67800009769873271668837921834, 5.62598324945398400124070368527, 6.37186269168781413932399400089, 7.19424759975230597984108332942, 8.311809411868578166377794605677, 9.059688423120075490001155956337, 9.522316586323420987453871900079, 10.54211486161353107634328925086
