| L(s) = 1 | − 0.422·2-s + 3-s − 1.82·4-s − 3.08·5-s − 0.422·6-s − 4.30·7-s + 1.61·8-s + 9-s + 1.30·10-s − 2.58·11-s − 1.82·12-s − 1.28·13-s + 1.81·14-s − 3.08·15-s + 2.95·16-s + 17-s − 0.422·18-s − 3.31·19-s + 5.61·20-s − 4.30·21-s + 1.09·22-s + 0.151·23-s + 1.61·24-s + 4.50·25-s + 0.544·26-s + 27-s + 7.83·28-s + ⋯ |
| L(s) = 1 | − 0.298·2-s + 0.577·3-s − 0.910·4-s − 1.37·5-s − 0.172·6-s − 1.62·7-s + 0.571·8-s + 0.333·9-s + 0.412·10-s − 0.779·11-s − 0.525·12-s − 0.356·13-s + 0.486·14-s − 0.795·15-s + 0.739·16-s + 0.242·17-s − 0.0996·18-s − 0.759·19-s + 1.25·20-s − 0.938·21-s + 0.233·22-s + 0.0315·23-s + 0.329·24-s + 0.900·25-s + 0.106·26-s + 0.192·27-s + 1.48·28-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 | \( 1 + 0.422T + 2T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 19 | \( 1 + 3.31T + 19T^{2} \) |
| 23 | \( 1 - 0.151T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 - 2.00T + 31T^{2} \) |
| 37 | \( 1 - 4.30T + 37T^{2} \) |
| 41 | \( 1 - 8.90T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 6.67T + 47T^{2} \) |
| 53 | \( 1 - 7.99T + 53T^{2} \) |
| 59 | \( 1 - 0.837T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 - 6.68T + 67T^{2} \) |
| 71 | \( 1 - 7.71T + 71T^{2} \) |
| 73 | \( 1 - 7.41T + 73T^{2} \) |
| 79 | \( 1 + 8.32T + 79T^{2} \) |
| 83 | \( 1 + 15.3T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 - 6.86T + 97T^{2} \) |
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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
−7.66159500004499823668719419697, −7.07284687948035166390208551944, −6.19742286909222422109249958177, −5.31705491398759935621448230664, −4.25500130438196892536695533417, −4.04043435365413809632525841978, −3.13430650975989730882345042804, −2.54853589424322544434939707814, −0.802342565801363973609993411931, 0,
0.802342565801363973609993411931, 2.54853589424322544434939707814, 3.13430650975989730882345042804, 4.04043435365413809632525841978, 4.25500130438196892536695533417, 5.31705491398759935621448230664, 6.19742286909222422109249958177, 7.07284687948035166390208551944, 7.66159500004499823668719419697
