| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s + 24-s + 25-s + 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s + 12-s + 13-s + 15-s + 16-s + 17-s + 18-s − 19-s + 20-s + 23-s + 24-s + 25-s + 26-s + 27-s + 29-s + 30-s − 31-s + 32-s + 34-s + 36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(12.20311641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.20311641\) |
| \(L(1)\) |
\(\approx\) |
\(3.930997600\) |
| \(L(1)\) |
\(\approx\) |
\(3.930997600\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 43 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.89958244816572675931697466473, −18.10030824811745903913073632931, −17.161963620614579573754304134607, −16.40903246378085965839953013824, −15.779308391430808663439213714822, −14.88537354783603253827742022208, −14.47935966038212527974894712413, −13.83415016363256614869008975802, −13.18630056974355958256876406755, −12.78457338716709414714039741691, −11.95579107252190787736229359966, −10.691347502339440273868588521062, −10.510212781325477215632429525672, −9.4613634712732409766724680610, −8.755424969317741587311564916553, −7.97945599149392240620246587269, −7.05898409745157298770809056335, −6.43355140879226242773636473198, −5.6657228902958230743713398771, −4.869429575378504637887620119018, −4.01376815913069282809486524456, −3.22183238037248455931263033543, −2.63526107769581095377306982312, −1.70032797615674983900354550778, −1.16564981916145965924873759221,
1.16564981916145965924873759221, 1.70032797615674983900354550778, 2.63526107769581095377306982312, 3.22183238037248455931263033543, 4.01376815913069282809486524456, 4.869429575378504637887620119018, 5.6657228902958230743713398771, 6.43355140879226242773636473198, 7.05898409745157298770809056335, 7.97945599149392240620246587269, 8.755424969317741587311564916553, 9.4613634712732409766724680610, 10.510212781325477215632429525672, 10.691347502339440273868588521062, 11.95579107252190787736229359966, 12.78457338716709414714039741691, 13.18630056974355958256876406755, 13.83415016363256614869008975802, 14.47935966038212527974894712413, 14.88537354783603253827742022208, 15.779308391430808663439213714822, 16.40903246378085965839953013824, 17.161963620614579573754304134607, 18.10030824811745903913073632931, 18.89958244816572675931697466473
