| L(s) = 1 | + 1.04·2-s − 3-s − 0.901·4-s + 2.25·5-s − 1.04·6-s − 3.56·7-s − 3.04·8-s + 9-s + 2.36·10-s − 0.421·11-s + 0.901·12-s + 1.27·13-s − 3.73·14-s − 2.25·15-s − 1.38·16-s − 17-s + 1.04·18-s + 7.63·19-s − 2.03·20-s + 3.56·21-s − 0.441·22-s − 3.06·23-s + 3.04·24-s + 0.0948·25-s + 1.33·26-s − 27-s + 3.21·28-s + ⋯ |
| L(s) = 1 | + 0.741·2-s − 0.577·3-s − 0.450·4-s + 1.00·5-s − 0.427·6-s − 1.34·7-s − 1.07·8-s + 0.333·9-s + 0.748·10-s − 0.127·11-s + 0.260·12-s + 0.352·13-s − 0.998·14-s − 0.582·15-s − 0.345·16-s − 0.242·17-s + 0.247·18-s + 1.75·19-s − 0.455·20-s + 0.778·21-s − 0.0941·22-s − 0.640·23-s + 0.620·24-s + 0.0189·25-s + 0.261·26-s − 0.192·27-s + 0.607·28-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 | \( 1 - 1.04T + 2T^{2} \) |
| 5 | \( 1 - 2.25T + 5T^{2} \) |
| 7 | \( 1 + 3.56T + 7T^{2} \) |
| 11 | \( 1 + 0.421T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 19 | \( 1 - 7.63T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 6.05T + 29T^{2} \) |
| 31 | \( 1 - 0.390T + 31T^{2} \) |
| 37 | \( 1 + 8.69T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 - 4.23T + 43T^{2} \) |
| 47 | \( 1 + 4.45T + 47T^{2} \) |
| 53 | \( 1 - 5.96T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 3.31T + 61T^{2} \) |
| 67 | \( 1 + 9.42T + 67T^{2} \) |
| 71 | \( 1 + 3.49T + 71T^{2} \) |
| 73 | \( 1 + 2.53T + 73T^{2} \) |
| 79 | \( 1 - 3.30T + 79T^{2} \) |
| 83 | \( 1 - 7.63T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.10692131375270747473616966724, −6.57252694789037672684781158124, −5.89300401513794177939116911490, −5.50781047659943164272603495003, −4.85756758275427168805430845602, −3.85895307033183082570398064181, −3.27981458355845013863521528206, −2.46939440459064785503337548325, −1.16920194707153782200621743969, 0,
1.16920194707153782200621743969, 2.46939440459064785503337548325, 3.27981458355845013863521528206, 3.85895307033183082570398064181, 4.85756758275427168805430845602, 5.50781047659943164272603495003, 5.89300401513794177939116911490, 6.57252694789037672684781158124, 7.10692131375270747473616966724
