| L(s) = 1 | − 2.65·2-s − 2.58·3-s + 5.07·4-s + 0.599·5-s + 6.87·6-s − 3.90·7-s − 8.17·8-s + 3.67·9-s − 1.59·10-s − 2.24·11-s − 13.1·12-s + 13-s + 10.3·14-s − 1.54·15-s + 11.6·16-s − 5.38·17-s − 9.76·18-s − 1.97·19-s + 3.04·20-s + 10.0·21-s + 5.96·22-s + 6.80·23-s + 21.1·24-s − 4.64·25-s − 2.65·26-s − 1.73·27-s − 19.8·28-s + ⋯ |
| L(s) = 1 | − 1.88·2-s − 1.49·3-s + 2.53·4-s + 0.268·5-s + 2.80·6-s − 1.47·7-s − 2.89·8-s + 1.22·9-s − 0.504·10-s − 0.676·11-s − 3.78·12-s + 0.277·13-s + 2.77·14-s − 0.399·15-s + 2.90·16-s − 1.30·17-s − 2.30·18-s − 0.452·19-s + 0.680·20-s + 2.20·21-s + 1.27·22-s + 1.41·23-s + 4.31·24-s − 0.928·25-s − 0.521·26-s − 0.333·27-s − 3.74·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 + 2.58T + 3T^{2} \) |
| 5 | \( 1 - 0.599T + 5T^{2} \) |
| 7 | \( 1 + 3.90T + 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 19 | \( 1 + 1.97T + 19T^{2} \) |
| 23 | \( 1 - 6.80T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 - 1.00T + 31T^{2} \) |
| 37 | \( 1 + 0.135T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 2.44T + 43T^{2} \) |
| 47 | \( 1 + 7.37T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 5.66T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 + 8.28T + 67T^{2} \) |
| 71 | \( 1 - 5.77T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 - 9.98T + 83T^{2} \) |
| 89 | \( 1 - 0.0258T + 89T^{2} \) |
| 97 | \( 1 + 8.17T + 97T^{2} \) |
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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
−7.38659465368981256383899120146, −6.78566508449271362415801360429, −6.33625062563799631587178629134, −5.85902928248299977673485015703, −5.01039453054262082839448283808, −3.72051037704930043580033433355, −2.69636437323222573111975153382, −1.87631436296697441402941560421, −0.65344389828480060914112398326, 0,
0.65344389828480060914112398326, 1.87631436296697441402941560421, 2.69636437323222573111975153382, 3.72051037704930043580033433355, 5.01039453054262082839448283808, 5.85902928248299977673485015703, 6.33625062563799631587178629134, 6.78566508449271362415801360429, 7.38659465368981256383899120146
