| L(s) = 1 | + 2-s − 1.52·3-s + 4-s − 3.77·5-s − 1.52·6-s + 0.908·7-s + 8-s − 0.662·9-s − 3.77·10-s − 2.19·11-s − 1.52·12-s − 1.27·13-s + 0.908·14-s + 5.76·15-s + 16-s + 0.0603·17-s − 0.662·18-s + 4.24·19-s − 3.77·20-s − 1.38·21-s − 2.19·22-s − 1.92·23-s − 1.52·24-s + 9.23·25-s − 1.27·26-s + 5.59·27-s + 0.908·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.882·3-s + 0.5·4-s − 1.68·5-s − 0.624·6-s + 0.343·7-s + 0.353·8-s − 0.220·9-s − 1.19·10-s − 0.663·11-s − 0.441·12-s − 0.353·13-s + 0.242·14-s + 1.48·15-s + 0.250·16-s + 0.0146·17-s − 0.156·18-s + 0.974·19-s − 0.843·20-s − 0.303·21-s − 0.468·22-s − 0.400·23-s − 0.312·24-s + 1.84·25-s − 0.250·26-s + 1.07·27-s + 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3011 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.52T + 3T^{2} \) |
| 5 | \( 1 + 3.77T + 5T^{2} \) |
| 7 | \( 1 - 0.908T + 7T^{2} \) |
| 11 | \( 1 + 2.19T + 11T^{2} \) |
| 13 | \( 1 + 1.27T + 13T^{2} \) |
| 17 | \( 1 - 0.0603T + 17T^{2} \) |
| 19 | \( 1 - 4.24T + 19T^{2} \) |
| 23 | \( 1 + 1.92T + 23T^{2} \) |
| 29 | \( 1 - 9.79T + 29T^{2} \) |
| 31 | \( 1 - 7.27T + 31T^{2} \) |
| 37 | \( 1 + 6.02T + 37T^{2} \) |
| 41 | \( 1 + 4.25T + 41T^{2} \) |
| 43 | \( 1 - 12.3T + 43T^{2} \) |
| 47 | \( 1 + 5.29T + 47T^{2} \) |
| 53 | \( 1 + 1.73T + 53T^{2} \) |
| 59 | \( 1 + 5.60T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 0.803T + 67T^{2} \) |
| 71 | \( 1 + 4.81T + 71T^{2} \) |
| 73 | \( 1 - 2.23T + 73T^{2} \) |
| 79 | \( 1 - 8.73T + 79T^{2} \) |
| 83 | \( 1 - 1.24T + 83T^{2} \) |
| 89 | \( 1 + 3.82T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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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
−7.85327305387819263489173539210, −6.88715490121313326242519576183, −6.31518587101905230360532294439, −5.32672623871660423609609111000, −4.82895458778545741992760904800, −4.29265522300270978655510344508, −3.26274514653644394637213746237, −2.71197444998794107290849304174, −1.07442116710462641063714059938, 0,
1.07442116710462641063714059938, 2.71197444998794107290849304174, 3.26274514653644394637213746237, 4.29265522300270978655510344508, 4.82895458778545741992760904800, 5.32672623871660423609609111000, 6.31518587101905230360532294439, 6.88715490121313326242519576183, 7.85327305387819263489173539210
