| L(s) = 1 | + 2-s + 2.23·3-s + 4-s + 0.607·5-s + 2.23·6-s − 1.94·7-s + 8-s + 2.00·9-s + 0.607·10-s − 1.03·11-s + 2.23·12-s − 4.02·13-s − 1.94·14-s + 1.35·15-s + 16-s − 3.02·17-s + 2.00·18-s + 19-s + 0.607·20-s − 4.36·21-s − 1.03·22-s − 4.16·23-s + 2.23·24-s − 4.63·25-s − 4.02·26-s − 2.22·27-s − 1.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.29·3-s + 0.5·4-s + 0.271·5-s + 0.913·6-s − 0.736·7-s + 0.353·8-s + 0.668·9-s + 0.191·10-s − 0.312·11-s + 0.645·12-s − 1.11·13-s − 0.520·14-s + 0.350·15-s + 0.250·16-s − 0.733·17-s + 0.472·18-s + 0.229·19-s + 0.135·20-s − 0.951·21-s − 0.221·22-s − 0.868·23-s + 0.456·24-s − 0.926·25-s − 0.788·26-s − 0.428·27-s − 0.368·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 5 | \( 1 - 0.607T + 5T^{2} \) |
| 7 | \( 1 + 1.94T + 7T^{2} \) |
| 11 | \( 1 + 1.03T + 11T^{2} \) |
| 13 | \( 1 + 4.02T + 13T^{2} \) |
| 17 | \( 1 + 3.02T + 17T^{2} \) |
| 23 | \( 1 + 4.16T + 23T^{2} \) |
| 29 | \( 1 - 0.615T + 29T^{2} \) |
| 31 | \( 1 + 9.20T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 0.126T + 41T^{2} \) |
| 43 | \( 1 - 5.11T + 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 - 3.69T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 + 5.49T + 61T^{2} \) |
| 67 | \( 1 + 5.99T + 67T^{2} \) |
| 71 | \( 1 + 6.59T + 71T^{2} \) |
| 73 | \( 1 - 6.60T + 73T^{2} \) |
| 79 | \( 1 - 0.650T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 4.78T + 89T^{2} \) |
| 97 | \( 1 - 0.168T + 97T^{2} \) |
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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
−7.49176351060178272838441389361, −6.91545399200286288939564465358, −5.98722681629676099198501506134, −5.45921719617154474285548074448, −4.42527073319741075655929266374, −3.83693006281422663596729435621, −3.04579666434937690250060394677, −2.42132995695630793489246353393, −1.84619456252831327977844355175, 0,
1.84619456252831327977844355175, 2.42132995695630793489246353393, 3.04579666434937690250060394677, 3.83693006281422663596729435621, 4.42527073319741075655929266374, 5.45921719617154474285548074448, 5.98722681629676099198501506134, 6.91545399200286288939564465358, 7.49176351060178272838441389361
