| L(s) = 1 | − 2-s + 2.07·3-s + 4-s − 0.312·5-s − 2.07·6-s + 0.578·7-s − 8-s + 1.31·9-s + 0.312·10-s − 1.72·11-s + 2.07·12-s − 4.32·13-s − 0.578·14-s − 0.650·15-s + 16-s + 1.57·17-s − 1.31·18-s + 6.45·19-s − 0.312·20-s + 1.20·21-s + 1.72·22-s − 0.697·23-s − 2.07·24-s − 4.90·25-s + 4.32·26-s − 3.49·27-s + 0.578·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.19·3-s + 0.5·4-s − 0.139·5-s − 0.848·6-s + 0.218·7-s − 0.353·8-s + 0.439·9-s + 0.0989·10-s − 0.520·11-s + 0.599·12-s − 1.19·13-s − 0.154·14-s − 0.167·15-s + 0.250·16-s + 0.382·17-s − 0.311·18-s + 1.48·19-s − 0.0699·20-s + 0.262·21-s + 0.368·22-s − 0.145·23-s − 0.424·24-s − 0.980·25-s + 0.847·26-s − 0.672·27-s + 0.109·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 2.07T + 3T^{2} \) |
| 5 | \( 1 + 0.312T + 5T^{2} \) |
| 7 | \( 1 - 0.578T + 7T^{2} \) |
| 11 | \( 1 + 1.72T + 11T^{2} \) |
| 13 | \( 1 + 4.32T + 13T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 - 6.45T + 19T^{2} \) |
| 23 | \( 1 + 0.697T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 37 | \( 1 - 0.997T + 37T^{2} \) |
| 41 | \( 1 + 1.46T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 1.94T + 47T^{2} \) |
| 53 | \( 1 - 0.438T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 + 8.60T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 - 0.00609T + 71T^{2} \) |
| 73 | \( 1 - 14.5T + 73T^{2} \) |
| 79 | \( 1 - 9.47T + 79T^{2} \) |
| 83 | \( 1 + 2.75T + 83T^{2} \) |
| 89 | \( 1 + 7.60T + 89T^{2} \) |
| 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
−7.922258505501359855934122417947, −7.41285361715638004347437313365, −6.59998677339414669929708582931, −5.50495236020984466534848848112, −4.89927811568434461286557451408, −3.71885343533644830236663984055, −3.02704312917306497226517368972, −2.36843624406840728080785720234, −1.46834922798654081111875099237, 0,
1.46834922798654081111875099237, 2.36843624406840728080785720234, 3.02704312917306497226517368972, 3.71885343533644830236663984055, 4.89927811568434461286557451408, 5.50495236020984466534848848112, 6.59998677339414669929708582931, 7.41285361715638004347437313365, 7.922258505501359855934122417947
