| L(s) = 1 | − 1.86·2-s − 1.84·3-s + 1.46·4-s − 5-s + 3.42·6-s + 3.02·7-s + 0.990·8-s + 0.389·9-s + 1.86·10-s + 2.15·11-s − 2.70·12-s − 4.82·13-s − 5.62·14-s + 1.84·15-s − 4.78·16-s + 4.55·17-s − 0.724·18-s − 2.01·19-s − 1.46·20-s − 5.56·21-s − 4.01·22-s − 4.26·23-s − 1.82·24-s + 25-s + 8.99·26-s + 4.80·27-s + 4.43·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s − 1.06·3-s + 0.734·4-s − 0.447·5-s + 1.39·6-s + 1.14·7-s + 0.350·8-s + 0.129·9-s + 0.588·10-s + 0.649·11-s − 0.780·12-s − 1.33·13-s − 1.50·14-s + 0.475·15-s − 1.19·16-s + 1.10·17-s − 0.170·18-s − 0.462·19-s − 0.328·20-s − 1.21·21-s − 0.855·22-s − 0.889·23-s − 0.372·24-s + 0.200·25-s + 1.76·26-s + 0.925·27-s + 0.838·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 3 | \( 1 + 1.84T + 3T^{2} \) |
| 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 2.15T + 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 4.26T + 23T^{2} \) |
| 29 | \( 1 + 9.85T + 29T^{2} \) |
| 31 | \( 1 - 6.31T + 31T^{2} \) |
| 37 | \( 1 - 0.983T + 37T^{2} \) |
| 41 | \( 1 + 6.83T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 + 0.00928T + 47T^{2} \) |
| 53 | \( 1 - 3.87T + 53T^{2} \) |
| 59 | \( 1 - 6.42T + 59T^{2} \) |
| 61 | \( 1 - 5.10T + 61T^{2} \) |
| 67 | \( 1 - 5.89T + 67T^{2} \) |
| 71 | \( 1 - 3.71T + 71T^{2} \) |
| 73 | \( 1 - 3.05T + 73T^{2} \) |
| 79 | \( 1 + 4.62T + 79T^{2} \) |
| 83 | \( 1 + 2.34T + 83T^{2} \) |
| 89 | \( 1 + 4.99T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.671611111237779475555088981837, −8.016569063468775257601636174068, −7.44711825802367536262694463617, −6.64523954863191150628978559753, −5.53141350751754127721233864036, −4.86410312229680415615060342630, −3.97256875053378073709213154292, −2.25546560278081839174482669000, −1.15383322795255142563654115579, 0,
1.15383322795255142563654115579, 2.25546560278081839174482669000, 3.97256875053378073709213154292, 4.86410312229680415615060342630, 5.53141350751754127721233864036, 6.64523954863191150628978559753, 7.44711825802367536262694463617, 8.016569063468775257601636174068, 8.671611111237779475555088981837
