L(s) = 1 | − 2-s − 0.286·3-s + 4-s + 3.05·5-s + 0.286·6-s − 7-s − 8-s − 2.91·9-s − 3.05·10-s − 1.07·11-s − 0.286·12-s + 0.416·13-s + 14-s − 0.876·15-s + 16-s + 3.47·17-s + 2.91·18-s − 8.09·19-s + 3.05·20-s + 0.286·21-s + 1.07·22-s + 1.29·23-s + 0.286·24-s + 4.35·25-s − 0.416·26-s + 1.69·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.165·3-s + 0.5·4-s + 1.36·5-s + 0.116·6-s − 0.377·7-s − 0.353·8-s − 0.972·9-s − 0.967·10-s − 0.324·11-s − 0.0826·12-s + 0.115·13-s + 0.267·14-s − 0.226·15-s + 0.250·16-s + 0.841·17-s + 0.687·18-s − 1.85·19-s + 0.683·20-s + 0.0625·21-s + 0.229·22-s + 0.270·23-s + 0.0584·24-s + 0.871·25-s − 0.0816·26-s + 0.326·27-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 | \( 1 + 0.286T + 3T^{2} \) |
| 5 | \( 1 - 3.05T + 5T^{2} \) |
| 11 | \( 1 + 1.07T + 11T^{2} \) |
| 13 | \( 1 - 0.416T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 1.29T + 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 - 0.387T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 + 1.60T + 43T^{2} \) |
| 47 | \( 1 - 0.853T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 3.69T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 7.58T + 67T^{2} \) |
| 71 | \( 1 - 10.5T + 71T^{2} \) |
| 73 | \( 1 + 1.99T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 5.26T + 83T^{2} \) |
| 89 | \( 1 + 9.10T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−7.84892301694262409160855319977, −6.88247562623931528963041029726, −6.30189932527749548601253310483, −5.73017074570907231567412335892, −5.18349943558813232006314453165, −3.95433321437759480976405005952, −2.80853250418792692173131630273, −2.33628262649904247133992061764, −1.29619976795849948340652520042, 0,
1.29619976795849948340652520042, 2.33628262649904247133992061764, 2.80853250418792692173131630273, 3.95433321437759480976405005952, 5.18349943558813232006314453165, 5.73017074570907231567412335892, 6.30189932527749548601253310483, 6.88247562623931528963041029726, 7.84892301694262409160855319977