| L(s) = 1 | − 0.251·2-s + 3-s − 1.93·4-s + 2.63·5-s − 0.251·6-s + 1.22·7-s + 0.988·8-s + 9-s − 0.662·10-s − 0.402·11-s − 1.93·12-s − 0.940·13-s − 0.306·14-s + 2.63·15-s + 3.62·16-s − 17-s − 0.251·18-s + 1.25·19-s − 5.10·20-s + 1.22·21-s + 0.101·22-s − 8.86·23-s + 0.988·24-s + 1.94·25-s + 0.236·26-s + 27-s − 2.36·28-s + ⋯ |
| L(s) = 1 | − 0.177·2-s + 0.577·3-s − 0.968·4-s + 1.17·5-s − 0.102·6-s + 0.461·7-s + 0.349·8-s + 0.333·9-s − 0.209·10-s − 0.121·11-s − 0.559·12-s − 0.260·13-s − 0.0819·14-s + 0.680·15-s + 0.906·16-s − 0.242·17-s − 0.0591·18-s + 0.288·19-s − 1.14·20-s + 0.266·21-s + 0.0215·22-s − 1.84·23-s + 0.201·24-s + 0.389·25-s + 0.0463·26-s + 0.192·27-s − 0.446·28-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 | \( 1 + 0.251T + 2T^{2} \) |
| 5 | \( 1 - 2.63T + 5T^{2} \) |
| 7 | \( 1 - 1.22T + 7T^{2} \) |
| 11 | \( 1 + 0.402T + 11T^{2} \) |
| 13 | \( 1 + 0.940T + 13T^{2} \) |
| 19 | \( 1 - 1.25T + 19T^{2} \) |
| 23 | \( 1 + 8.86T + 23T^{2} \) |
| 29 | \( 1 - 1.22T + 29T^{2} \) |
| 31 | \( 1 + 6.62T + 31T^{2} \) |
| 37 | \( 1 - 2.87T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 + 7.04T + 43T^{2} \) |
| 47 | \( 1 + 2.20T + 47T^{2} \) |
| 53 | \( 1 + 4.59T + 53T^{2} \) |
| 59 | \( 1 + 6.65T + 59T^{2} \) |
| 61 | \( 1 - 4.50T + 61T^{2} \) |
| 67 | \( 1 + 0.159T + 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 3.73T + 73T^{2} \) |
| 79 | \( 1 - 6.14T + 79T^{2} \) |
| 83 | \( 1 + 14.4T + 83T^{2} \) |
| 89 | \( 1 + 0.974T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−7.78886990102954564855829152673, −6.82803037081648086423658745853, −5.97680927520449513324877553408, −5.37182981844869122324269543028, −4.68932857747846213748168715847, −3.95200585602432033145297165370, −3.08748392396160959199929151468, −2.00476714670813446305752586081, −1.51701475169946186890728211837, 0,
1.51701475169946186890728211837, 2.00476714670813446305752586081, 3.08748392396160959199929151468, 3.95200585602432033145297165370, 4.68932857747846213748168715847, 5.37182981844869122324269543028, 5.97680927520449513324877553408, 6.82803037081648086423658745853, 7.78886990102954564855829152673
