L(s) = 1 | + 2-s − 0.618·3-s + 4-s − 5-s − 0.618·6-s − 7-s + 8-s − 2.61·9-s − 10-s − 0.618·12-s + 0.763·13-s − 14-s + 0.618·15-s + 16-s − 0.381·17-s − 2.61·18-s − 1.14·19-s − 20-s + 0.618·21-s + 8.47·23-s − 0.618·24-s + 25-s + 0.763·26-s + 3.47·27-s − 28-s + 4.47·29-s + 0.618·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.356·3-s + 0.5·4-s − 0.447·5-s − 0.252·6-s − 0.377·7-s + 0.353·8-s − 0.872·9-s − 0.316·10-s − 0.178·12-s + 0.211·13-s − 0.267·14-s + 0.159·15-s + 0.250·16-s − 0.0926·17-s − 0.617·18-s − 0.262·19-s − 0.223·20-s + 0.134·21-s + 1.76·23-s − 0.126·24-s + 0.200·25-s + 0.149·26-s + 0.668·27-s − 0.188·28-s + 0.830·29-s + 0.112·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 0.618T + 3T^{2} \) |
| 13 | \( 1 - 0.763T + 13T^{2} \) |
| 17 | \( 1 + 0.381T + 17T^{2} \) |
| 19 | \( 1 + 1.14T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 + 6.76T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 9.09T + 43T^{2} \) |
| 47 | \( 1 - 9.23T + 47T^{2} \) |
| 53 | \( 1 + 7.23T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 0.291T + 61T^{2} \) |
| 67 | \( 1 + 5.61T + 67T^{2} \) |
| 71 | \( 1 + 6.94T + 71T^{2} \) |
| 73 | \( 1 - 3.85T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 2.14T + 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.11440971368456220158833147383, −6.79420639345199537114131175660, −5.95421010264292866288691172793, −5.32921937029041689622823744249, −4.75199285412731442987775992575, −3.83421448712248309622671149413, −3.16814649926131606829754012212, −2.51356037878663668023591568732, −1.23339180419163550408097559753, 0,
1.23339180419163550408097559753, 2.51356037878663668023591568732, 3.16814649926131606829754012212, 3.83421448712248309622671149413, 4.75199285412731442987775992575, 5.32921937029041689622823744249, 5.95421010264292866288691172793, 6.79420639345199537114131175660, 7.11440971368456220158833147383