| L(s) = 1 | − 2.64·2-s + 0.666·3-s + 5.00·4-s + 2.47·5-s − 1.76·6-s + 1.01·7-s − 7.94·8-s − 2.55·9-s − 6.55·10-s − 5.17·11-s + 3.33·12-s − 1.29·13-s − 2.67·14-s + 1.65·15-s + 11.0·16-s − 0.735·17-s + 6.76·18-s + 5.77·19-s + 12.3·20-s + 0.674·21-s + 13.7·22-s − 0.896·23-s − 5.29·24-s + 1.14·25-s + 3.41·26-s − 3.70·27-s + 5.06·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.384·3-s + 2.50·4-s + 1.10·5-s − 0.719·6-s + 0.382·7-s − 2.80·8-s − 0.851·9-s − 2.07·10-s − 1.56·11-s + 0.962·12-s − 0.358·13-s − 0.715·14-s + 0.426·15-s + 2.75·16-s − 0.178·17-s + 1.59·18-s + 1.32·19-s + 2.77·20-s + 0.147·21-s + 2.92·22-s − 0.186·23-s − 1.08·24-s + 0.228·25-s + 0.670·26-s − 0.712·27-s + 0.956·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 + 2.64T + 2T^{2} \) |
| 3 | \( 1 - 0.666T + 3T^{2} \) |
| 5 | \( 1 - 2.47T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 5.17T + 11T^{2} \) |
| 13 | \( 1 + 1.29T + 13T^{2} \) |
| 17 | \( 1 + 0.735T + 17T^{2} \) |
| 19 | \( 1 - 5.77T + 19T^{2} \) |
| 23 | \( 1 + 0.896T + 23T^{2} \) |
| 29 | \( 1 + 3.42T + 29T^{2} \) |
| 31 | \( 1 - 3.87T + 31T^{2} \) |
| 37 | \( 1 + 4.52T + 37T^{2} \) |
| 41 | \( 1 + 10.4T + 41T^{2} \) |
| 47 | \( 1 + 0.494T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 1.40T + 59T^{2} \) |
| 61 | \( 1 + 2.40T + 61T^{2} \) |
| 67 | \( 1 + 5.71T + 67T^{2} \) |
| 71 | \( 1 - 0.277T + 71T^{2} \) |
| 73 | \( 1 + 7.80T + 73T^{2} \) |
| 79 | \( 1 - 9.25T + 79T^{2} \) |
| 83 | \( 1 + 6.32T + 83T^{2} \) |
| 89 | \( 1 + 5.33T + 89T^{2} \) |
| 97 | \( 1 + 6.49T + 97T^{2} \) |
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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
−8.855957194746507907404936558699, −8.199635077704198935704693872085, −7.64818331651688911986706108439, −6.81635539179437603620814178023, −5.73678641133839539582514644048, −5.23342611027039851154163334170, −3.08645448695517302986141165631, −2.43204775512500475946094688549, −1.58662580499389094685105018310, 0,
1.58662580499389094685105018310, 2.43204775512500475946094688549, 3.08645448695517302986141165631, 5.23342611027039851154163334170, 5.73678641133839539582514644048, 6.81635539179437603620814178023, 7.64818331651688911986706108439, 8.199635077704198935704693872085, 8.855957194746507907404936558699
