| L(s) = 1 | + 0.587·2-s − 2.68·3-s − 1.65·4-s + 5-s − 1.57·6-s + 7-s − 2.14·8-s + 4.18·9-s + 0.587·10-s + 5.23·11-s + 4.43·12-s + 4.08·13-s + 0.587·14-s − 2.68·15-s + 2.04·16-s − 0.816·17-s + 2.46·18-s − 6.53·19-s − 1.65·20-s − 2.68·21-s + 3.07·22-s − 2.03·23-s + 5.75·24-s + 25-s + 2.39·26-s − 3.18·27-s − 1.65·28-s + ⋯ |
| L(s) = 1 | + 0.415·2-s − 1.54·3-s − 0.827·4-s + 0.447·5-s − 0.643·6-s + 0.377·7-s − 0.759·8-s + 1.39·9-s + 0.185·10-s + 1.57·11-s + 1.28·12-s + 1.13·13-s + 0.157·14-s − 0.692·15-s + 0.511·16-s − 0.197·17-s + 0.580·18-s − 1.49·19-s − 0.369·20-s − 0.585·21-s + 0.656·22-s − 0.424·23-s + 1.17·24-s + 0.200·25-s + 0.470·26-s − 0.613·27-s − 0.312·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 - 0.587T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 11 | \( 1 - 5.23T + 11T^{2} \) |
| 13 | \( 1 - 4.08T + 13T^{2} \) |
| 17 | \( 1 + 0.816T + 17T^{2} \) |
| 19 | \( 1 + 6.53T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 5.67T + 29T^{2} \) |
| 31 | \( 1 + 4.66T + 31T^{2} \) |
| 37 | \( 1 + 5.17T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 1.97T + 43T^{2} \) |
| 47 | \( 1 - 5.79T + 47T^{2} \) |
| 53 | \( 1 + 9.73T + 53T^{2} \) |
| 59 | \( 1 - 5.16T + 59T^{2} \) |
| 61 | \( 1 - 9.73T + 61T^{2} \) |
| 67 | \( 1 - 0.415T + 67T^{2} \) |
| 71 | \( 1 + 5.99T + 71T^{2} \) |
| 73 | \( 1 + 4.99T + 73T^{2} \) |
| 79 | \( 1 - 1.91T + 79T^{2} \) |
| 83 | \( 1 + 4.91T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−7.13465193632808998494470252368, −6.47713422176554349311359610115, −5.96033134824066920194257173752, −5.54781888930973791577614781591, −4.68778933616497158075434735105, −4.09144055697663595179744770126, −3.58181572814045038886201074928, −1.91237489576705948761901685480, −1.12128632545358607326486080351, 0,
1.12128632545358607326486080351, 1.91237489576705948761901685480, 3.58181572814045038886201074928, 4.09144055697663595179744770126, 4.68778933616497158075434735105, 5.54781888930973791577614781591, 5.96033134824066920194257173752, 6.47713422176554349311359610115, 7.13465193632808998494470252368
