| L(s) = 1 | + 1.95·2-s + 3-s + 1.82·4-s + 0.269·5-s + 1.95·6-s − 1.92·7-s − 0.339·8-s + 9-s + 0.527·10-s − 1.66·11-s + 1.82·12-s + 3.03·13-s − 3.75·14-s + 0.269·15-s − 4.31·16-s + 17-s + 1.95·18-s + 1.51·19-s + 0.492·20-s − 1.92·21-s − 3.25·22-s − 7.30·23-s − 0.339·24-s − 4.92·25-s + 5.94·26-s + 27-s − 3.50·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s + 0.577·3-s + 0.913·4-s + 0.120·5-s + 0.798·6-s − 0.725·7-s − 0.120·8-s + 0.333·9-s + 0.166·10-s − 0.502·11-s + 0.527·12-s + 0.843·13-s − 1.00·14-s + 0.0696·15-s − 1.07·16-s + 0.242·17-s + 0.461·18-s + 0.347·19-s + 0.110·20-s − 0.419·21-s − 0.694·22-s − 1.52·23-s − 0.0693·24-s − 0.985·25-s + 1.16·26-s + 0.192·27-s − 0.662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 - 1.95T + 2T^{2} \) |
| 5 | \( 1 - 0.269T + 5T^{2} \) |
| 7 | \( 1 + 1.92T + 7T^{2} \) |
| 11 | \( 1 + 1.66T + 11T^{2} \) |
| 13 | \( 1 - 3.03T + 13T^{2} \) |
| 19 | \( 1 - 1.51T + 19T^{2} \) |
| 23 | \( 1 + 7.30T + 23T^{2} \) |
| 29 | \( 1 + 8.83T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 9.04T + 37T^{2} \) |
| 41 | \( 1 - 7.21T + 41T^{2} \) |
| 43 | \( 1 - 7.52T + 43T^{2} \) |
| 47 | \( 1 + 3.28T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 + 9.96T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 - 5.74T + 67T^{2} \) |
| 71 | \( 1 - 3.78T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 + 14.5T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.51839393956639895789076641677, −6.50918217747900957122578567528, −5.79202585294432082490982375460, −5.65244435241397524947515869357, −4.38678038153782804251231730859, −3.92960340519574327954068978090, −3.29514061000062592286182840420, −2.57195522349371406334302092289, −1.70131463597358396602572037083, 0,
1.70131463597358396602572037083, 2.57195522349371406334302092289, 3.29514061000062592286182840420, 3.92960340519574327954068978090, 4.38678038153782804251231730859, 5.65244435241397524947515869357, 5.79202585294432082490982375460, 6.50918217747900957122578567528, 7.51839393956639895789076641677
