| L(s) = 1 | + 2-s + 4-s + 3.73·5-s + 2.73·7-s + 8-s + 3.73·10-s + 1.26·11-s + 2.73·14-s + 16-s + 5.73·17-s − 4.73·19-s + 3.73·20-s + 1.26·22-s − 4.19·23-s + 8.92·25-s + 2.73·28-s + 4.46·29-s − 1.46·31-s + 32-s + 5.73·34-s + 10.1·35-s − 3.53·37-s − 4.73·38-s + 3.73·40-s − 9.39·41-s − 9.66·43-s + 1.26·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.66·5-s + 1.03·7-s + 0.353·8-s + 1.18·10-s + 0.382·11-s + 0.730·14-s + 0.250·16-s + 1.39·17-s − 1.08·19-s + 0.834·20-s + 0.270·22-s − 0.874·23-s + 1.78·25-s + 0.516·28-s + 0.828·29-s − 0.262·31-s + 0.176·32-s + 0.983·34-s + 1.72·35-s − 0.581·37-s − 0.767·38-s + 0.590·40-s − 1.46·41-s − 1.47·43-s + 0.191·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.727992212\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.727992212\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.73T + 5T^{2} \) |
| 7 | \( 1 - 2.73T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 + 4.73T + 19T^{2} \) |
| 23 | \( 1 + 4.19T + 23T^{2} \) |
| 29 | \( 1 - 4.46T + 29T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + 3.53T + 37T^{2} \) |
| 41 | \( 1 + 9.39T + 41T^{2} \) |
| 43 | \( 1 + 9.66T + 43T^{2} \) |
| 47 | \( 1 - 2.19T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 + 9.19T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 4.73T + 71T^{2} \) |
| 73 | \( 1 - 6.26T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 + 0.196T + 83T^{2} \) |
| 89 | \( 1 + 9.46T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.617049658119660568483925116553, −8.047840735048900369522183051577, −6.92462666217734711604906139166, −6.30846835238485380523293235764, −5.52630091215188444648591721997, −5.07401177267264750177802321262, −4.12387342501864834954304991178, −3.01976516942970441412326113851, −1.96037030203415569551932536120, −1.44980365953219590467071226800,
1.44980365953219590467071226800, 1.96037030203415569551932536120, 3.01976516942970441412326113851, 4.12387342501864834954304991178, 5.07401177267264750177802321262, 5.52630091215188444648591721997, 6.30846835238485380523293235764, 6.92462666217734711604906139166, 8.047840735048900369522183051577, 8.617049658119660568483925116553
