| L(s) = 1 | − 0.833·2-s − 3-s − 1.30·4-s − 2.56·5-s + 0.833·6-s + 1.99·7-s + 2.75·8-s + 9-s + 2.13·10-s + 2.12·11-s + 1.30·12-s + 4.16·13-s − 1.66·14-s + 2.56·15-s + 0.316·16-s + 17-s − 0.833·18-s − 4.35·19-s + 3.34·20-s − 1.99·21-s − 1.76·22-s + 4.76·23-s − 2.75·24-s + 1.56·25-s − 3.47·26-s − 27-s − 2.60·28-s + ⋯ |
| L(s) = 1 | − 0.589·2-s − 0.577·3-s − 0.652·4-s − 1.14·5-s + 0.340·6-s + 0.754·7-s + 0.973·8-s + 0.333·9-s + 0.674·10-s + 0.640·11-s + 0.376·12-s + 1.15·13-s − 0.444·14-s + 0.661·15-s + 0.0791·16-s + 0.242·17-s − 0.196·18-s − 0.998·19-s + 0.748·20-s − 0.435·21-s − 0.377·22-s + 0.994·23-s − 0.562·24-s + 0.312·25-s − 0.680·26-s − 0.192·27-s − 0.492·28-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 | \( 1 + 0.833T + 2T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 7 | \( 1 - 1.99T + 7T^{2} \) |
| 11 | \( 1 - 2.12T + 11T^{2} \) |
| 13 | \( 1 - 4.16T + 13T^{2} \) |
| 19 | \( 1 + 4.35T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 1.37T + 29T^{2} \) |
| 31 | \( 1 + 5.24T + 31T^{2} \) |
| 37 | \( 1 + 0.204T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 - 0.0932T + 43T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 + 4.98T + 53T^{2} \) |
| 59 | \( 1 - 6.46T + 59T^{2} \) |
| 61 | \( 1 + 5.18T + 61T^{2} \) |
| 67 | \( 1 - 4.51T + 67T^{2} \) |
| 71 | \( 1 - 4.87T + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 9.63T + 79T^{2} \) |
| 83 | \( 1 + 7.95T + 83T^{2} \) |
| 89 | \( 1 - 9.99T + 89T^{2} \) |
| 97 | \( 1 + 2.78T + 97T^{2} \) |
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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
−7.65190527496851189629988717860, −6.92248187312060560958252491570, −6.22441613295088675709312873066, −5.17704177826207612155716505920, −4.70722431842179650736833572890, −3.89750562423695257824728342541, −3.48732567183830484153933489164, −1.77605024178296572294060582556, −1.03409644555026339738558708222, 0,
1.03409644555026339738558708222, 1.77605024178296572294060582556, 3.48732567183830484153933489164, 3.89750562423695257824728342541, 4.70722431842179650736833572890, 5.17704177826207612155716505920, 6.22441613295088675709312873066, 6.92248187312060560958252491570, 7.65190527496851189629988717860
