| L(s) = 1 | + 0.294·2-s − 1.11·3-s − 1.91·4-s − 5-s − 0.327·6-s − 7-s − 1.15·8-s − 1.76·9-s − 0.294·10-s − 2.87·11-s + 2.12·12-s + 1.17·13-s − 0.294·14-s + 1.11·15-s + 3.48·16-s − 2.91·17-s − 0.519·18-s − 8.14·19-s + 1.91·20-s + 1.11·21-s − 0.845·22-s + 6.04·23-s + 1.28·24-s + 25-s + 0.345·26-s + 5.29·27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.208·2-s − 0.641·3-s − 0.956·4-s − 0.447·5-s − 0.133·6-s − 0.377·7-s − 0.407·8-s − 0.588·9-s − 0.0930·10-s − 0.866·11-s + 0.613·12-s + 0.325·13-s − 0.0786·14-s + 0.286·15-s + 0.871·16-s − 0.706·17-s − 0.122·18-s − 1.86·19-s + 0.427·20-s + 0.242·21-s − 0.180·22-s + 1.25·23-s + 0.261·24-s + 0.200·25-s + 0.0677·26-s + 1.01·27-s + 0.361·28-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 | \( 1 - 0.294T + 2T^{2} \) |
| 3 | \( 1 + 1.11T + 3T^{2} \) |
| 11 | \( 1 + 2.87T + 11T^{2} \) |
| 13 | \( 1 - 1.17T + 13T^{2} \) |
| 17 | \( 1 + 2.91T + 17T^{2} \) |
| 19 | \( 1 + 8.14T + 19T^{2} \) |
| 23 | \( 1 - 6.04T + 23T^{2} \) |
| 29 | \( 1 - 8.39T + 29T^{2} \) |
| 31 | \( 1 + 8.38T + 31T^{2} \) |
| 37 | \( 1 - 6.89T + 37T^{2} \) |
| 41 | \( 1 - 2.64T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 - 7.92T + 47T^{2} \) |
| 53 | \( 1 - 1.13T + 53T^{2} \) |
| 59 | \( 1 + 9.00T + 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 2.49T + 67T^{2} \) |
| 71 | \( 1 + 0.912T + 71T^{2} \) |
| 73 | \( 1 + 1.05T + 73T^{2} \) |
| 79 | \( 1 - 3.65T + 79T^{2} \) |
| 83 | \( 1 - 5.73T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
| 97 | \( 1 + 6.23T + 97T^{2} \) |
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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
−7.52589777498690326001854851670, −6.60563139224876260757430178401, −6.00222722994884601179318769467, −5.40671531305797584638165205529, −4.55188485973091368085396515799, −4.21460114405118458318128138562, −3.12705792913186323300655069080, −2.46331679396569759908724274585, −0.807468058285421863324265758748, 0,
0.807468058285421863324265758748, 2.46331679396569759908724274585, 3.12705792913186323300655069080, 4.21460114405118458318128138562, 4.55188485973091368085396515799, 5.40671531305797584638165205529, 6.00222722994884601179318769467, 6.60563139224876260757430178401, 7.52589777498690326001854851670
