| L(s) = 1 | − 2.15·2-s + 3-s + 2.64·4-s − 0.239·5-s − 2.15·6-s + 7-s − 1.40·8-s + 9-s + 0.515·10-s − 3.99·11-s + 2.64·12-s − 1.61·13-s − 2.15·14-s − 0.239·15-s − 2.27·16-s + 4.61·17-s − 2.15·18-s − 4.17·19-s − 0.633·20-s + 21-s + 8.62·22-s + 5.22·23-s − 1.40·24-s − 4.94·25-s + 3.48·26-s + 27-s + 2.64·28-s + ⋯ |
| L(s) = 1 | − 1.52·2-s + 0.577·3-s + 1.32·4-s − 0.106·5-s − 0.880·6-s + 0.377·7-s − 0.495·8-s + 0.333·9-s + 0.163·10-s − 1.20·11-s + 0.764·12-s − 0.448·13-s − 0.576·14-s − 0.0617·15-s − 0.569·16-s + 1.11·17-s − 0.508·18-s − 0.957·19-s − 0.141·20-s + 0.218·21-s + 1.83·22-s + 1.08·23-s − 0.285·24-s − 0.988·25-s + 0.683·26-s + 0.192·27-s + 0.500·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 + 2.15T + 2T^{2} \) |
| 5 | \( 1 + 0.239T + 5T^{2} \) |
| 11 | \( 1 + 3.99T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 4.61T + 17T^{2} \) |
| 19 | \( 1 + 4.17T + 19T^{2} \) |
| 23 | \( 1 - 5.22T + 23T^{2} \) |
| 29 | \( 1 - 9.10T + 29T^{2} \) |
| 31 | \( 1 + 6.57T + 31T^{2} \) |
| 37 | \( 1 + 0.353T + 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 7.33T + 43T^{2} \) |
| 47 | \( 1 - 3.52T + 47T^{2} \) |
| 53 | \( 1 + 0.655T + 53T^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 + 7.05T + 61T^{2} \) |
| 67 | \( 1 + 0.167T + 67T^{2} \) |
| 71 | \( 1 - 6.98T + 71T^{2} \) |
| 73 | \( 1 + 2.79T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 6.75T + 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 1.79T + 97T^{2} \) |
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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
−8.366341346804682346408885673434, −7.977685499377859733412580498619, −7.36761416680151942248353610019, −6.58902676554451421343333300555, −5.34825248132872967815931881445, −4.56100039758419152895686563053, −3.23898604671392589707376902050, −2.35067960140450992167580650730, −1.40355511455004245193478482159, 0,
1.40355511455004245193478482159, 2.35067960140450992167580650730, 3.23898604671392589707376902050, 4.56100039758419152895686563053, 5.34825248132872967815931881445, 6.58902676554451421343333300555, 7.36761416680151942248353610019, 7.977685499377859733412580498619, 8.366341346804682346408885673434
