| L(s) = 1 | − 2-s + 1.06·3-s + 4-s + 4.09·5-s − 1.06·6-s + 2.34·7-s − 8-s − 1.87·9-s − 4.09·10-s + 4.82·11-s + 1.06·12-s + 4.41·13-s − 2.34·14-s + 4.34·15-s + 16-s − 2.51·17-s + 1.87·18-s − 2.07·19-s + 4.09·20-s + 2.48·21-s − 4.82·22-s − 2.02·23-s − 1.06·24-s + 11.7·25-s − 4.41·26-s − 5.17·27-s + 2.34·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.613·3-s + 0.5·4-s + 1.83·5-s − 0.433·6-s + 0.884·7-s − 0.353·8-s − 0.623·9-s − 1.29·10-s + 1.45·11-s + 0.306·12-s + 1.22·13-s − 0.625·14-s + 1.12·15-s + 0.250·16-s − 0.610·17-s + 0.440·18-s − 0.475·19-s + 0.915·20-s + 0.542·21-s − 1.02·22-s − 0.422·23-s − 0.216·24-s + 2.35·25-s − 0.865·26-s − 0.996·27-s + 0.442·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1682 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.501765394\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.501765394\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 - 1.06T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.41T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 + 2.02T + 23T^{2} \) |
| 31 | \( 1 + 0.814T + 31T^{2} \) |
| 37 | \( 1 + 2.59T + 37T^{2} \) |
| 41 | \( 1 + 0.104T + 41T^{2} \) |
| 43 | \( 1 + 3.15T + 43T^{2} \) |
| 47 | \( 1 - 9.52T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 8.95T + 59T^{2} \) |
| 61 | \( 1 + 5.91T + 61T^{2} \) |
| 67 | \( 1 - 5.20T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 + 2.52T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 - 2.97T + 97T^{2} \) |
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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
−9.163002687463500800288237073493, −8.769904717326455762859776031095, −8.154032336845731691846016441662, −6.85503286865435433744028845280, −6.18384106879801150699757592217, −5.60161122176826588973030658521, −4.29493377915966498163619498522, −3.04770878685415660995750810099, −1.92169550579335618061690580143, −1.44226286578634658557135139241,
1.44226286578634658557135139241, 1.92169550579335618061690580143, 3.04770878685415660995750810099, 4.29493377915966498163619498522, 5.60161122176826588973030658521, 6.18384106879801150699757592217, 6.85503286865435433744028845280, 8.154032336845731691846016441662, 8.769904717326455762859776031095, 9.163002687463500800288237073493
