| L(s) = 1 | − 1.21·2-s − 3-s − 0.527·4-s + 1.21·6-s − 0.223·7-s + 3.06·8-s + 9-s + 1.53·11-s + 0.527·12-s − 5.64·13-s + 0.271·14-s − 2.66·16-s − 2.05·17-s − 1.21·18-s − 1.59·19-s + 0.223·21-s − 1.86·22-s + 7.37·23-s − 3.06·24-s + 6.84·26-s − 27-s + 0.118·28-s + 2.56·29-s + 0.459·31-s − 2.89·32-s − 1.53·33-s + 2.49·34-s + ⋯ |
| L(s) = 1 | − 0.857·2-s − 0.577·3-s − 0.263·4-s + 0.495·6-s − 0.0845·7-s + 1.08·8-s + 0.333·9-s + 0.463·11-s + 0.152·12-s − 1.56·13-s + 0.0725·14-s − 0.666·16-s − 0.498·17-s − 0.285·18-s − 0.365·19-s + 0.0488·21-s − 0.397·22-s + 1.53·23-s − 0.626·24-s + 1.34·26-s − 0.192·27-s + 0.0223·28-s + 0.476·29-s + 0.0824·31-s − 0.512·32-s − 0.267·33-s + 0.427·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5676611366\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5676611366\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 + 1.21T + 2T^{2} \) |
| 7 | \( 1 + 0.223T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 5.64T + 13T^{2} \) |
| 17 | \( 1 + 2.05T + 17T^{2} \) |
| 19 | \( 1 + 1.59T + 19T^{2} \) |
| 23 | \( 1 - 7.37T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 - 0.459T + 31T^{2} \) |
| 37 | \( 1 + 0.00801T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 5.66T + 43T^{2} \) |
| 47 | \( 1 + 8.59T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 5.11T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 5.57T + 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 - 0.407T + 89T^{2} \) |
| 97 | \( 1 - 3.40T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.424018607373566155123469176601, −7.68326891307846965292828821454, −6.94206253646291559063328904242, −6.46400520498091192324376390331, −5.11664447407411858715202041803, −4.87054833916634187648625849435, −3.95041586241684976592849048031, −2.72381363920950055992334604559, −1.61882800866723556282534576979, −0.51075270922621825837368866143,
0.51075270922621825837368866143, 1.61882800866723556282534576979, 2.72381363920950055992334604559, 3.95041586241684976592849048031, 4.87054833916634187648625849435, 5.11664447407411858715202041803, 6.46400520498091192324376390331, 6.94206253646291559063328904242, 7.68326891307846965292828821454, 8.424018607373566155123469176601
