| L(s) = 1 | − 2-s − 0.937·3-s + 4-s + 2.71·5-s + 0.937·6-s + 7-s − 8-s − 2.12·9-s − 2.71·10-s + 1.93·11-s − 0.937·12-s − 3.51·13-s − 14-s − 2.54·15-s + 16-s − 5.52·17-s + 2.12·18-s − 4.16·19-s + 2.71·20-s − 0.937·21-s − 1.93·22-s + 4.00·23-s + 0.937·24-s + 2.36·25-s + 3.51·26-s + 4.80·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.541·3-s + 0.5·4-s + 1.21·5-s + 0.382·6-s + 0.377·7-s − 0.353·8-s − 0.706·9-s − 0.858·10-s + 0.582·11-s − 0.270·12-s − 0.975·13-s − 0.267·14-s − 0.656·15-s + 0.250·16-s − 1.34·17-s + 0.499·18-s − 0.955·19-s + 0.606·20-s − 0.204·21-s − 0.411·22-s + 0.836·23-s + 0.191·24-s + 0.472·25-s + 0.689·26-s + 0.924·27-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.937T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 11 | \( 1 - 1.93T + 11T^{2} \) |
| 13 | \( 1 + 3.51T + 13T^{2} \) |
| 17 | \( 1 + 5.52T + 17T^{2} \) |
| 19 | \( 1 + 4.16T + 19T^{2} \) |
| 23 | \( 1 - 4.00T + 23T^{2} \) |
| 29 | \( 1 + 1.55T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 3.87T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 8.67T + 43T^{2} \) |
| 47 | \( 1 - 4.13T + 47T^{2} \) |
| 53 | \( 1 + 8.52T + 53T^{2} \) |
| 59 | \( 1 - 1.76T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 + 8.75T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 - 3.70T + 79T^{2} \) |
| 83 | \( 1 - 1.50T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.74616758419976705463927131271, −6.89957233538115269619969946422, −6.28425870403230234349124893662, −5.82509599697980876479137136363, −4.94708633239008205386479637971, −4.25908697978621332661399259403, −2.72484782505891760239223553808, −2.28657277340902176507591591448, −1.26504825058349272654331109255, 0,
1.26504825058349272654331109255, 2.28657277340902176507591591448, 2.72484782505891760239223553808, 4.25908697978621332661399259403, 4.94708633239008205386479637971, 5.82509599697980876479137136363, 6.28425870403230234349124893662, 6.89957233538115269619969946422, 7.74616758419976705463927131271
