| L(s) = 1 | − 2.49·2-s − 3.32·3-s + 4.23·4-s − 4.15·5-s + 8.30·6-s − 2.74·7-s − 5.58·8-s + 8.06·9-s + 10.3·10-s + 5.53·11-s − 14.0·12-s + 4.58·13-s + 6.85·14-s + 13.8·15-s + 5.47·16-s + 0.508·17-s − 20.1·18-s − 0.582·19-s − 17.6·20-s + 9.13·21-s − 13.8·22-s − 1.53·23-s + 18.5·24-s + 12.2·25-s − 11.4·26-s − 16.8·27-s − 11.6·28-s + ⋯ |
| L(s) = 1 | − 1.76·2-s − 1.92·3-s + 2.11·4-s − 1.85·5-s + 3.39·6-s − 1.03·7-s − 1.97·8-s + 2.68·9-s + 3.28·10-s + 1.66·11-s − 4.06·12-s + 1.27·13-s + 1.83·14-s + 3.57·15-s + 1.36·16-s + 0.123·17-s − 4.74·18-s − 0.133·19-s − 3.93·20-s + 1.99·21-s − 2.94·22-s − 0.320·23-s + 3.79·24-s + 2.45·25-s − 2.24·26-s − 3.24·27-s − 2.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 71 | \( 1 + T \) |
| 113 | \( 1 + T \) |
| good | 2 | \( 1 + 2.49T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 + 4.15T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 5.53T + 11T^{2} \) |
| 13 | \( 1 - 4.58T + 13T^{2} \) |
| 17 | \( 1 - 0.508T + 17T^{2} \) |
| 19 | \( 1 + 0.582T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 6.73T + 31T^{2} \) |
| 37 | \( 1 - 3.94T + 37T^{2} \) |
| 41 | \( 1 + 0.699T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 - 8.69T + 47T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 - 7.59T + 61T^{2} \) |
| 67 | \( 1 + 12.3T + 67T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 + 12.6T + 89T^{2} \) |
| 97 | \( 1 - 4.50T + 97T^{2} \) |
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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
−7.41914062923641818399473406189, −6.85696406410579616974732507683, −6.32456013226300380076046228994, −6.00576903911602323617941114528, −4.43644365658634335860434651096, −4.07391730453814493117017907750, −3.12638749662962546144995970510, −1.29968031077455588937423073919, −0.851406460308151085586252268775, 0,
0.851406460308151085586252268775, 1.29968031077455588937423073919, 3.12638749662962546144995970510, 4.07391730453814493117017907750, 4.43644365658634335860434651096, 6.00576903911602323617941114528, 6.32456013226300380076046228994, 6.85696406410579616974732507683, 7.41914062923641818399473406189
