| L(s) = 1 | − 0.943·2-s + 3-s − 1.10·4-s + 3.50·5-s − 0.943·6-s − 2.32·7-s + 2.93·8-s + 9-s − 3.31·10-s − 5.01·11-s − 1.10·12-s + 1.00·13-s + 2.19·14-s + 3.50·15-s − 0.551·16-s − 17-s − 0.943·18-s + 3.26·19-s − 3.89·20-s − 2.32·21-s + 4.73·22-s − 3.55·23-s + 2.93·24-s + 7.31·25-s − 0.948·26-s + 27-s + 2.58·28-s + ⋯ |
| L(s) = 1 | − 0.667·2-s + 0.577·3-s − 0.554·4-s + 1.56·5-s − 0.385·6-s − 0.879·7-s + 1.03·8-s + 0.333·9-s − 1.04·10-s − 1.51·11-s − 0.320·12-s + 0.278·13-s + 0.587·14-s + 0.905·15-s − 0.137·16-s − 0.242·17-s − 0.222·18-s + 0.749·19-s − 0.870·20-s − 0.508·21-s + 1.00·22-s − 0.740·23-s + 0.599·24-s + 1.46·25-s − 0.186·26-s + 0.192·27-s + 0.487·28-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 | \( 1 + 0.943T + 2T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 + 2.32T + 7T^{2} \) |
| 11 | \( 1 + 5.01T + 11T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 + 4.86T + 29T^{2} \) |
| 31 | \( 1 - 2.30T + 31T^{2} \) |
| 37 | \( 1 + 1.16T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.68T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 5.16T + 59T^{2} \) |
| 61 | \( 1 + 1.49T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 6.84T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 8.66T + 83T^{2} \) |
| 89 | \( 1 + 16.3T + 89T^{2} \) |
| 97 | \( 1 + 6.59T + 97T^{2} \) |
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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
−7.56071447367216260853250845756, −7.02096857948676843956782227191, −5.92976350236946294056795027935, −5.58021388525945989020539079224, −4.76646398781632102957670603375, −3.79478947326828372248557857544, −2.85259075618264400506224882854, −2.21192542393273280662814814892, −1.28527487009159258968087057841, 0,
1.28527487009159258968087057841, 2.21192542393273280662814814892, 2.85259075618264400506224882854, 3.79478947326828372248557857544, 4.76646398781632102957670603375, 5.58021388525945989020539079224, 5.92976350236946294056795027935, 7.02096857948676843956782227191, 7.56071447367216260853250845756
