| L(s) = 1 | − 2.47·2-s + 3.30·3-s + 4.14·4-s + 5-s − 8.18·6-s + 4.80·7-s − 5.31·8-s + 7.89·9-s − 2.47·10-s + 3.29·11-s + 13.6·12-s − 3.61·13-s − 11.9·14-s + 3.30·15-s + 4.88·16-s + 4.59·17-s − 19.5·18-s − 4.43·19-s + 4.14·20-s + 15.8·21-s − 8.15·22-s − 2.29·23-s − 17.5·24-s + 25-s + 8.95·26-s + 16.1·27-s + 19.9·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 1.90·3-s + 2.07·4-s + 0.447·5-s − 3.34·6-s + 1.81·7-s − 1.87·8-s + 2.63·9-s − 0.783·10-s + 0.992·11-s + 3.94·12-s − 1.00·13-s − 3.18·14-s + 0.852·15-s + 1.22·16-s + 1.11·17-s − 4.61·18-s − 1.01·19-s + 0.926·20-s + 3.45·21-s − 1.73·22-s − 0.479·23-s − 3.58·24-s + 0.200·25-s + 1.75·26-s + 3.11·27-s + 3.76·28-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
| \(L(1)\) |
\(\approx\) |
\(2.241587202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.241587202\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.47T + 2T^{2} \) |
| 3 | \( 1 - 3.30T + 3T^{2} \) |
| 7 | \( 1 - 4.80T + 7T^{2} \) |
| 11 | \( 1 - 3.29T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 4.59T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 7.40T + 31T^{2} \) |
| 37 | \( 1 + 8.94T + 37T^{2} \) |
| 41 | \( 1 + 2.65T + 41T^{2} \) |
| 43 | \( 1 - 6.14T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 8.83T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 + 1.15T + 67T^{2} \) |
| 71 | \( 1 + 0.742T + 71T^{2} \) |
| 73 | \( 1 - 0.903T + 73T^{2} \) |
| 79 | \( 1 + 8.69T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 8.20T + 97T^{2} \) |
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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
−8.903962279540133244767585614881, −8.553927863714893796256983791958, −7.85020003448312190109115650529, −7.43233169316097674872326347781, −6.59956526164546196137965979585, −5.01844762221518961050812258626, −3.97150533876855911790934475751, −2.71782340433300250956952516743, −1.77655093074173960331120925337, −1.49974899840956773091441094583,
1.49974899840956773091441094583, 1.77655093074173960331120925337, 2.71782340433300250956952516743, 3.97150533876855911790934475751, 5.01844762221518961050812258626, 6.59956526164546196137965979585, 7.43233169316097674872326347781, 7.85020003448312190109115650529, 8.553927863714893796256983791958, 8.903962279540133244767585614881
