| L(s) = 1 | + 0.460·2-s + 3-s − 1.78·4-s − 2.47·5-s + 0.460·6-s − 1.36·7-s − 1.74·8-s + 9-s − 1.13·10-s − 5.81·11-s − 1.78·12-s + 4.84·13-s − 0.630·14-s − 2.47·15-s + 2.77·16-s + 17-s + 0.460·18-s + 1.61·19-s + 4.42·20-s − 1.36·21-s − 2.67·22-s − 0.293·23-s − 1.74·24-s + 1.13·25-s + 2.22·26-s + 27-s + 2.44·28-s + ⋯ |
| L(s) = 1 | + 0.325·2-s + 0.577·3-s − 0.894·4-s − 1.10·5-s + 0.187·6-s − 0.517·7-s − 0.616·8-s + 0.333·9-s − 0.360·10-s − 1.75·11-s − 0.516·12-s + 1.34·13-s − 0.168·14-s − 0.639·15-s + 0.693·16-s + 0.242·17-s + 0.108·18-s + 0.371·19-s + 0.990·20-s − 0.298·21-s − 0.570·22-s − 0.0611·23-s − 0.355·24-s + 0.226·25-s + 0.437·26-s + 0.192·27-s + 0.462·28-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 | \( 1 - 0.460T + 2T^{2} \) |
| 5 | \( 1 + 2.47T + 5T^{2} \) |
| 7 | \( 1 + 1.36T + 7T^{2} \) |
| 11 | \( 1 + 5.81T + 11T^{2} \) |
| 13 | \( 1 - 4.84T + 13T^{2} \) |
| 19 | \( 1 - 1.61T + 19T^{2} \) |
| 23 | \( 1 + 0.293T + 23T^{2} \) |
| 29 | \( 1 - 3.39T + 29T^{2} \) |
| 31 | \( 1 - 8.89T + 31T^{2} \) |
| 37 | \( 1 - 7.21T + 37T^{2} \) |
| 41 | \( 1 + 6.29T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 + 8.98T + 47T^{2} \) |
| 53 | \( 1 + 8.32T + 53T^{2} \) |
| 59 | \( 1 - 8.47T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 - 2.20T + 67T^{2} \) |
| 71 | \( 1 - 4.16T + 71T^{2} \) |
| 73 | \( 1 - 15.0T + 73T^{2} \) |
| 79 | \( 1 + 1.76T + 79T^{2} \) |
| 83 | \( 1 + 9.95T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 6.55T + 97T^{2} \) |
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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
−7.86634719563831147661984291350, −6.80709604795585581488278976269, −6.06123595657415085330853072334, −5.19158150674148242684274250217, −4.59091883557164738125868793044, −3.80306402155978762374835661034, −3.26188775173491687080817609666, −2.64593672976813120986643532878, −1.03892147840954433896332110309, 0,
1.03892147840954433896332110309, 2.64593672976813120986643532878, 3.26188775173491687080817609666, 3.80306402155978762374835661034, 4.59091883557164738125868793044, 5.19158150674148242684274250217, 6.06123595657415085330853072334, 6.80709604795585581488278976269, 7.86634719563831147661984291350
