| L(s) = 1 | + 1.57i·2-s − 0.472·4-s + (1.72 + 1.42i)5-s − 1.87i·7-s + 2.40i·8-s + (−2.24 + 2.70i)10-s − 1.92·11-s + 0.248i·13-s + 2.95·14-s − 4.72·16-s + 7.06i·17-s − 19-s + (−0.813 − 0.672i)20-s − 3.02i·22-s + 1.20i·23-s + ⋯ |
| L(s) = 1 | + 1.11i·2-s − 0.236·4-s + (0.770 + 0.637i)5-s − 0.710i·7-s + 0.849i·8-s + (−0.708 + 0.856i)10-s − 0.579·11-s + 0.0688i·13-s + 0.789·14-s − 1.18·16-s + 1.71i·17-s − 0.229·19-s + (−0.181 − 0.150i)20-s − 0.643i·22-s + 0.250i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.602468 + 1.67420i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.602468 + 1.67420i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1.72 - 1.42i)T \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 1.57iT - 2T^{2} \) |
| 7 | \( 1 + 1.87iT - 7T^{2} \) |
| 11 | \( 1 + 1.92T + 11T^{2} \) |
| 13 | \( 1 - 0.248iT - 13T^{2} \) |
| 17 | \( 1 - 7.06iT - 17T^{2} \) |
| 23 | \( 1 - 1.20iT - 23T^{2} \) |
| 29 | \( 1 - 2.58T + 29T^{2} \) |
| 31 | \( 1 - 8.69T + 31T^{2} \) |
| 37 | \( 1 - 7.86iT - 37T^{2} \) |
| 41 | \( 1 + 1.52T + 41T^{2} \) |
| 43 | \( 1 - 4.15iT - 43T^{2} \) |
| 47 | \( 1 + 5.96iT - 47T^{2} \) |
| 53 | \( 1 + 7.11iT - 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 5.87T + 61T^{2} \) |
| 67 | \( 1 + 9.53iT - 67T^{2} \) |
| 71 | \( 1 - 9.53T + 71T^{2} \) |
| 73 | \( 1 + 6.69iT - 73T^{2} \) |
| 79 | \( 1 - 0.348T + 79T^{2} \) |
| 83 | \( 1 + 2.41iT - 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 7.70iT - 97T^{2} \) |
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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
−10.46801726897049601842731762927, −9.738926647831828317254423978228, −8.416978483408863794607620747978, −7.935417350907262444035513011119, −6.82900970278000736526443283652, −6.39693134753561606259803162793, −5.54650414147814967452721080149, −4.48963311626447906101580310248, −3.08111230277674266958760869416, −1.81487289617263923547742101067,
0.867078431670072747726241439742, 2.31429359148562095952995007176, 2.82925767864963353732391708099, 4.38871251955419402246286130893, 5.27304075499559096967427662980, 6.23086595600442730395522234577, 7.28091999953992204113400374714, 8.507112867833135809942986554315, 9.285835756818065963731677039451, 9.898549443933558803586145152418
