| L(s) = 1 | − 2-s + 2.69·3-s + 4-s − 1.10·5-s − 2.69·6-s − 4.42·7-s − 8-s + 4.25·9-s + 1.10·10-s + 3.71·11-s + 2.69·12-s + 2.61·13-s + 4.42·14-s − 2.96·15-s + 16-s + 4.26·17-s − 4.25·18-s − 6.00·19-s − 1.10·20-s − 11.9·21-s − 3.71·22-s + 6.59·23-s − 2.69·24-s − 3.78·25-s − 2.61·26-s + 3.36·27-s − 4.42·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.55·3-s + 0.5·4-s − 0.492·5-s − 1.09·6-s − 1.67·7-s − 0.353·8-s + 1.41·9-s + 0.348·10-s + 1.12·11-s + 0.777·12-s + 0.724·13-s + 1.18·14-s − 0.765·15-s + 0.250·16-s + 1.03·17-s − 1.00·18-s − 1.37·19-s − 0.246·20-s − 2.60·21-s − 0.792·22-s + 1.37·23-s − 0.549·24-s − 0.757·25-s − 0.512·26-s + 0.648·27-s − 0.836·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.969084506\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.969084506\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
| good | 3 | \( 1 - 2.69T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 + 4.42T + 7T^{2} \) |
| 11 | \( 1 - 3.71T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 6.00T + 19T^{2} \) |
| 23 | \( 1 - 6.59T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 - 0.953T + 31T^{2} \) |
| 37 | \( 1 - 9.19T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 - 5.63T + 47T^{2} \) |
| 53 | \( 1 - 7.62T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 5.87T + 67T^{2} \) |
| 71 | \( 1 - 15.8T + 71T^{2} \) |
| 73 | \( 1 + 1.44T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 + 8.44T + 83T^{2} \) |
| 89 | \( 1 - 17.8T + 89T^{2} \) |
| 97 | \( 1 + 6.87T + 97T^{2} \) |
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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
−8.600830017968121090154596555122, −7.909181322739110705366109044414, −7.09018310760391999112990889614, −6.58400167639068712274359174374, −5.78122383711090279712140945388, −4.08638204821137963795831908052, −3.62579717767406239036025650432, −3.02586766413836298116851863470, −2.07301991011388283278739790296, −0.828509475982995976897843029522,
0.828509475982995976897843029522, 2.07301991011388283278739790296, 3.02586766413836298116851863470, 3.62579717767406239036025650432, 4.08638204821137963795831908052, 5.78122383711090279712140945388, 6.58400167639068712274359174374, 7.09018310760391999112990889614, 7.909181322739110705366109044414, 8.600830017968121090154596555122
