| L(s) = 1 | − 1.03·2-s − 3-s − 0.926·4-s − 2.19·5-s + 1.03·6-s + 0.525·7-s + 3.03·8-s + 9-s + 2.27·10-s + 3.25·11-s + 0.926·12-s − 6.70·13-s − 0.544·14-s + 2.19·15-s − 1.28·16-s + 17-s − 1.03·18-s + 3.24·19-s + 2.03·20-s − 0.525·21-s − 3.36·22-s − 3.92·23-s − 3.03·24-s − 0.191·25-s + 6.94·26-s − 27-s − 0.486·28-s + ⋯ |
| L(s) = 1 | − 0.732·2-s − 0.577·3-s − 0.463·4-s − 0.980·5-s + 0.423·6-s + 0.198·7-s + 1.07·8-s + 0.333·9-s + 0.718·10-s + 0.980·11-s + 0.267·12-s − 1.85·13-s − 0.145·14-s + 0.566·15-s − 0.322·16-s + 0.242·17-s − 0.244·18-s + 0.745·19-s + 0.454·20-s − 0.114·21-s − 0.718·22-s − 0.819·23-s − 0.618·24-s − 0.0382·25-s + 1.36·26-s − 0.192·27-s − 0.0920·28-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 | \( 1 + 1.03T + 2T^{2} \) |
| 5 | \( 1 + 2.19T + 5T^{2} \) |
| 7 | \( 1 - 0.525T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 6.70T + 13T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 0.274T + 29T^{2} \) |
| 31 | \( 1 - 7.90T + 31T^{2} \) |
| 37 | \( 1 + 2.01T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 1.64T + 47T^{2} \) |
| 53 | \( 1 - 3.44T + 53T^{2} \) |
| 59 | \( 1 + 4.12T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 + 9.87T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 + 5.57T + 73T^{2} \) |
| 79 | \( 1 + 2.77T + 79T^{2} \) |
| 83 | \( 1 + 5.77T + 83T^{2} \) |
| 89 | \( 1 - 11.5T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−7.56236211795632656539372465962, −7.12846471163581752604116977121, −6.19202095632711602086179619264, −5.25230505583749131770921589166, −4.56975076853345972195462599271, −4.16419792802988378040726556064, −3.18833351920276168327359261301, −1.93035686417990286021174614285, −0.882309822276943552884774703790, 0,
0.882309822276943552884774703790, 1.93035686417990286021174614285, 3.18833351920276168327359261301, 4.16419792802988378040726556064, 4.56975076853345972195462599271, 5.25230505583749131770921589166, 6.19202095632711602086179619264, 7.12846471163581752604116977121, 7.56236211795632656539372465962
