| L(s) = 1 | − 2-s + 2.56·3-s + 4-s − 2.56·6-s − 5.12·7-s − 8-s + 3.56·9-s − 4·11-s + 2.56·12-s − 4.56·13-s + 5.12·14-s + 16-s − 17-s − 3.56·18-s − 2.56·19-s − 13.1·21-s + 4·22-s + 5.12·23-s − 2.56·24-s + 4.56·26-s + 1.43·27-s − 5.12·28-s − 5.68·29-s − 6.56·31-s − 32-s − 10.2·33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.47·3-s + 0.5·4-s − 1.04·6-s − 1.93·7-s − 0.353·8-s + 1.18·9-s − 1.20·11-s + 0.739·12-s − 1.26·13-s + 1.36·14-s + 0.250·16-s − 0.242·17-s − 0.839·18-s − 0.587·19-s − 2.86·21-s + 0.852·22-s + 1.06·23-s − 0.522·24-s + 0.894·26-s + 0.276·27-s − 0.968·28-s − 1.05·29-s − 1.17·31-s − 0.176·32-s − 1.78·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.56T + 13T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 23 | \( 1 - 5.12T + 23T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + 6.56T + 31T^{2} \) |
| 37 | \( 1 - 7.12T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 - 0.561T + 53T^{2} \) |
| 59 | \( 1 + 0.315T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 9.12T + 67T^{2} \) |
| 71 | \( 1 + 4.31T + 71T^{2} \) |
| 73 | \( 1 - 6.80T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.24T + 83T^{2} \) |
| 89 | \( 1 + 9.68T + 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−9.459940339775681491376327419231, −9.221513272123202818986154975250, −8.119395556067851899509936859992, −7.38374524006160531731326830467, −6.71714938948018178728954349000, −5.44977392067390631054451452685, −3.85348871141607991664510524975, −2.84924080126800190309222068587, −2.37985575272537982289991805366, 0,
2.37985575272537982289991805366, 2.84924080126800190309222068587, 3.85348871141607991664510524975, 5.44977392067390631054451452685, 6.71714938948018178728954349000, 7.38374524006160531731326830467, 8.119395556067851899509936859992, 9.221513272123202818986154975250, 9.459940339775681491376327419231
