L(s) = 1 | + 1.13·2-s − 0.715·4-s + 1.11·5-s + 7-s − 3.07·8-s + 1.26·10-s + 3.25·11-s + 6.02·13-s + 1.13·14-s − 2.05·16-s + 2.98·17-s − 6.12·19-s − 0.795·20-s + 3.68·22-s + 5.74·23-s − 3.76·25-s + 6.83·26-s − 0.715·28-s + 7.65·29-s + 7.30·31-s + 3.82·32-s + 3.37·34-s + 1.11·35-s + 5.93·37-s − 6.93·38-s − 3.42·40-s − 4.23·41-s + ⋯ |
L(s) = 1 | + 0.801·2-s − 0.357·4-s + 0.497·5-s + 0.377·7-s − 1.08·8-s + 0.398·10-s + 0.981·11-s + 1.67·13-s + 0.302·14-s − 0.514·16-s + 0.723·17-s − 1.40·19-s − 0.177·20-s + 0.786·22-s + 1.19·23-s − 0.752·25-s + 1.33·26-s − 0.135·28-s + 1.42·29-s + 1.31·31-s + 0.675·32-s + 0.579·34-s + 0.187·35-s + 0.974·37-s − 1.12·38-s − 0.540·40-s − 0.661·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.605452254\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.605452254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 - 1.13T + 2T^{2} \) |
| 5 | \( 1 - 1.11T + 5T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 - 6.02T + 13T^{2} \) |
| 17 | \( 1 - 2.98T + 17T^{2} \) |
| 19 | \( 1 + 6.12T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 - 5.93T + 37T^{2} \) |
| 41 | \( 1 + 4.23T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 4.13T + 53T^{2} \) |
| 59 | \( 1 - 7.82T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 + 1.91T + 67T^{2} \) |
| 71 | \( 1 + 1.98T + 71T^{2} \) |
| 73 | \( 1 - 0.455T + 73T^{2} \) |
| 79 | \( 1 - 8.00T + 79T^{2} \) |
| 83 | \( 1 + 4.44T + 83T^{2} \) |
| 89 | \( 1 + 6.16T + 89T^{2} \) |
| 97 | \( 1 - 3.77T + 97T^{2} \) |
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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
−8.135492690849437003598017860735, −6.64004178516637660377637216685, −6.41073452848138166751545411148, −5.79185883184722127093176432282, −4.87404773763450274698945676534, −4.39374421172901074882874691537, −3.58202545689994020022464505007, −2.98681445241741429147629794152, −1.73728324815464008841880666025, −0.897495756722945094052046660179,
0.897495756722945094052046660179, 1.73728324815464008841880666025, 2.98681445241741429147629794152, 3.58202545689994020022464505007, 4.39374421172901074882874691537, 4.87404773763450274698945676534, 5.79185883184722127093176432282, 6.41073452848138166751545411148, 6.64004178516637660377637216685, 8.135492690849437003598017860735