| L(s) = 1 | − 2-s − 1.69·3-s + 4-s + 3.76·5-s + 1.69·6-s − 7-s − 8-s − 0.129·9-s − 3.76·10-s + 5.13·11-s − 1.69·12-s + 2.53·13-s + 14-s − 6.37·15-s + 16-s + 1.63·17-s + 0.129·18-s + 2.24·19-s + 3.76·20-s + 1.69·21-s − 5.13·22-s + 4.75·23-s + 1.69·24-s + 9.17·25-s − 2.53·26-s + 5.30·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.978·3-s + 0.5·4-s + 1.68·5-s + 0.691·6-s − 0.377·7-s − 0.353·8-s − 0.0432·9-s − 1.19·10-s + 1.54·11-s − 0.489·12-s + 0.702·13-s + 0.267·14-s − 1.64·15-s + 0.250·16-s + 0.396·17-s + 0.0305·18-s + 0.514·19-s + 0.841·20-s + 0.369·21-s − 1.09·22-s + 0.990·23-s + 0.345·24-s + 1.83·25-s − 0.496·26-s + 1.02·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.742832669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.742832669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 1.69T + 3T^{2} \) |
| 5 | \( 1 - 3.76T + 5T^{2} \) |
| 11 | \( 1 - 5.13T + 11T^{2} \) |
| 13 | \( 1 - 2.53T + 13T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.75T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 + 6.10T + 41T^{2} \) |
| 43 | \( 1 + 7.09T + 43T^{2} \) |
| 47 | \( 1 - 1.29T + 47T^{2} \) |
| 53 | \( 1 - 1.88T + 53T^{2} \) |
| 59 | \( 1 - 7.44T + 59T^{2} \) |
| 61 | \( 1 + 0.00800T + 61T^{2} \) |
| 67 | \( 1 - 0.701T + 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 2.75T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 7.74T + 83T^{2} \) |
| 89 | \( 1 + 8.90T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.373272127351347084406694017250, −6.95588123168913940027752144987, −6.59129979760453737847868724744, −6.09470170625937521981661358392, −5.50617310832830917964166944607, −4.73377186279486115003617630146, −3.44530645884197015357206112001, −2.60877513025620116338970623092, −1.41193450770076352511436363706, −0.943225675075969223661102077749,
0.943225675075969223661102077749, 1.41193450770076352511436363706, 2.60877513025620116338970623092, 3.44530645884197015357206112001, 4.73377186279486115003617630146, 5.50617310832830917964166944607, 6.09470170625937521981661358392, 6.59129979760453737847868724744, 6.95588123168913940027752144987, 8.373272127351347084406694017250
