| L(s) = 1 | + 0.923·2-s − 3-s − 1.14·4-s + 1.71·5-s − 0.923·6-s + 3.15·7-s − 2.90·8-s + 9-s + 1.57·10-s − 2.69·11-s + 1.14·12-s − 2.09·13-s + 2.91·14-s − 1.71·15-s − 0.389·16-s + 17-s + 0.923·18-s + 2.82·19-s − 1.96·20-s − 3.15·21-s − 2.49·22-s − 1.04·23-s + 2.90·24-s − 2.07·25-s − 1.93·26-s − 27-s − 3.62·28-s + ⋯ |
| L(s) = 1 | + 0.652·2-s − 0.577·3-s − 0.573·4-s + 0.764·5-s − 0.376·6-s + 1.19·7-s − 1.02·8-s + 0.333·9-s + 0.499·10-s − 0.813·11-s + 0.331·12-s − 0.579·13-s + 0.779·14-s − 0.441·15-s − 0.0972·16-s + 0.242·17-s + 0.217·18-s + 0.649·19-s − 0.438·20-s − 0.688·21-s − 0.531·22-s − 0.218·23-s + 0.593·24-s − 0.415·25-s − 0.378·26-s − 0.192·27-s − 0.684·28-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 | \( 1 - 0.923T + 2T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 - 3.15T + 7T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 1.04T + 23T^{2} \) |
| 29 | \( 1 + 0.179T + 29T^{2} \) |
| 31 | \( 1 + 0.672T + 31T^{2} \) |
| 37 | \( 1 - 1.39T + 37T^{2} \) |
| 41 | \( 1 + 6.76T + 41T^{2} \) |
| 43 | \( 1 - 0.830T + 43T^{2} \) |
| 47 | \( 1 + 5.01T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 - 0.658T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 - 4.59T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 - 5.86T + 97T^{2} \) |
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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
−7.59028496445906855667456277194, −6.56415272626928290926524170139, −5.78930192220478408716059663139, −5.26311550604757280367663655276, −4.91295930162992325313474136889, −4.17563210980245212003718715075, −3.18015175858933267040487270645, −2.24246601043605138029621959357, −1.32111884274167789923689089111, 0,
1.32111884274167789923689089111, 2.24246601043605138029621959357, 3.18015175858933267040487270645, 4.17563210980245212003718715075, 4.91295930162992325313474136889, 5.26311550604757280367663655276, 5.78930192220478408716059663139, 6.56415272626928290926524170139, 7.59028496445906855667456277194
