| L(s) = 1 | + (1.65 − 0.506i)3-s − 5-s − 4.66i·7-s + (2.48 − 1.67i)9-s − 1.85·11-s + 5.81i·13-s + (−1.65 + 0.506i)15-s − 2.48i·17-s − 4.91·19-s + (−2.36 − 7.72i)21-s − 4.23i·23-s + 25-s + (3.27 − 4.03i)27-s + 2.15i·29-s − 6.26i·31-s + ⋯ |
| L(s) = 1 | + (0.956 − 0.292i)3-s − 0.447·5-s − 1.76i·7-s + (0.829 − 0.559i)9-s − 0.559·11-s + 1.61i·13-s + (−0.427 + 0.130i)15-s − 0.602i·17-s − 1.12·19-s + (−0.515 − 1.68i)21-s − 0.882i·23-s + 0.200·25-s + (0.629 − 0.777i)27-s + 0.399i·29-s − 1.12i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.309929254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.309929254\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.65 + 0.506i)T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + (8.18 + 0.228i)T \) |
| good | 7 | \( 1 + 4.66iT - 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 - 5.81iT - 13T^{2} \) |
| 17 | \( 1 + 2.48iT - 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 + 4.23iT - 23T^{2} \) |
| 29 | \( 1 - 2.15iT - 29T^{2} \) |
| 31 | \( 1 + 6.26iT - 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 4.35iT - 43T^{2} \) |
| 47 | \( 1 - 2.64iT - 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 8.02iT - 59T^{2} \) |
| 61 | \( 1 + 0.752iT - 61T^{2} \) |
| 71 | \( 1 + 7.41iT - 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 7.19iT - 79T^{2} \) |
| 83 | \( 1 + 12.4iT - 83T^{2} \) |
| 89 | \( 1 - 8.65iT - 89T^{2} \) |
| 97 | \( 1 + 0.0915iT - 97T^{2} \) |
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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
−8.003456451120372852283883811223, −7.32964352339858434754431777233, −6.97398151189919771745205197811, −6.21742537763525847597319508906, −4.53701353781086366094313666408, −4.34914986645684767399545694075, −3.55371249627164179986366080588, −2.51019308746650427031008423528, −1.53930035461614804838521248398, −0.30600326760625891040787948752,
1.68359434378314400219051331596, 2.67482074472303379408044855840, 3.12019288872619823697840310555, 4.10275089793109652376556718495, 5.16834923594473297091298076044, 5.60292100489808123902877529090, 6.58543148837584192112000934550, 7.69013531344633706358483385848, 8.181774381842571331749218011111, 8.641430820988135334428597101236
