| L(s) = 1 | − 1.50·2-s + 3-s + 0.255·4-s − 0.915·5-s − 1.50·6-s − 3.11·7-s + 2.61·8-s + 9-s + 1.37·10-s + 1.83·11-s + 0.255·12-s − 4.69·13-s + 4.67·14-s − 0.915·15-s − 4.44·16-s − 17-s − 1.50·18-s + 4.81·19-s − 0.233·20-s − 3.11·21-s − 2.75·22-s − 2.49·23-s + 2.61·24-s − 4.16·25-s + 7.05·26-s + 27-s − 0.795·28-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.577·3-s + 0.127·4-s − 0.409·5-s − 0.613·6-s − 1.17·7-s + 0.926·8-s + 0.333·9-s + 0.434·10-s + 0.552·11-s + 0.0737·12-s − 1.30·13-s + 1.24·14-s − 0.236·15-s − 1.11·16-s − 0.242·17-s − 0.353·18-s + 1.10·19-s − 0.0523·20-s − 0.679·21-s − 0.586·22-s − 0.520·23-s + 0.534·24-s − 0.832·25-s + 1.38·26-s + 0.192·27-s − 0.150·28-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 | \( 1 + 1.50T + 2T^{2} \) |
| 5 | \( 1 + 0.915T + 5T^{2} \) |
| 7 | \( 1 + 3.11T + 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 19 | \( 1 - 4.81T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 - 7.38T + 31T^{2} \) |
| 37 | \( 1 + 5.82T + 37T^{2} \) |
| 41 | \( 1 - 3.73T + 41T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 - 3.13T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 3.86T + 59T^{2} \) |
| 61 | \( 1 - 4.04T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 4.17T + 71T^{2} \) |
| 73 | \( 1 - 3.90T + 73T^{2} \) |
| 79 | \( 1 + 1.84T + 79T^{2} \) |
| 83 | \( 1 + 2.87T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−7.52023654785945207490989935430, −7.12283655989925926070419749981, −6.44591769677488425514028649133, −5.39415609935887084279679161754, −4.48248909477998272235684615051, −3.85080990213313800637943884830, −2.97701417156614626286596371914, −2.18103360127943163720284141678, −0.993350212763072398355099512923, 0,
0.993350212763072398355099512923, 2.18103360127943163720284141678, 2.97701417156614626286596371914, 3.85080990213313800637943884830, 4.48248909477998272235684615051, 5.39415609935887084279679161754, 6.44591769677488425514028649133, 7.12283655989925926070419749981, 7.52023654785945207490989935430
