L(s) = 1 | − 4.91·2-s − 7.41·3-s + 16.1·4-s + 5·5-s + 36.4·6-s + 29.5·7-s − 39.8·8-s + 28.0·9-s − 24.5·10-s − 11·11-s − 119.·12-s − 58.0·13-s − 145.·14-s − 37.0·15-s + 66.7·16-s + 78.5·17-s − 137.·18-s + 19·19-s + 80.5·20-s − 219.·21-s + 54.0·22-s + 154.·23-s + 295.·24-s + 25·25-s + 284.·26-s − 7.75·27-s + 476.·28-s + ⋯ |
L(s) = 1 | − 1.73·2-s − 1.42·3-s + 2.01·4-s + 0.447·5-s + 2.47·6-s + 1.59·7-s − 1.76·8-s + 1.03·9-s − 0.776·10-s − 0.301·11-s − 2.87·12-s − 1.23·13-s − 2.76·14-s − 0.638·15-s + 1.04·16-s + 1.12·17-s − 1.80·18-s + 0.229·19-s + 0.900·20-s − 2.27·21-s + 0.523·22-s + 1.40·23-s + 2.51·24-s + 0.200·25-s + 2.14·26-s − 0.0552·27-s + 3.21·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6837653262\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6837653262\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 5T \) |
| 11 | \( 1 + 11T \) |
| 19 | \( 1 - 19T \) |
good | 2 | \( 1 + 4.91T + 8T^{2} \) |
| 3 | \( 1 + 7.41T + 27T^{2} \) |
| 7 | \( 1 - 29.5T + 343T^{2} \) |
| 13 | \( 1 + 58.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 78.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 154.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 52.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 168.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 430.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 68.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 310.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 491.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 375.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 176.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 45.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 520.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 405.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 762.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 652.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.34e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.26e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 254.T + 9.12e5T^{2} \) |
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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
−9.649465342260983660693464750066, −8.817125646757025560550720334021, −7.68924676800510808601381565030, −7.43675540306470765851592682142, −6.30654655034413965750525685694, −5.31647800535557147901902226190, −4.78352572684092250172893484598, −2.57033956819625749359801695406, −1.42623141560295896873360950197, −0.67683853408712207040020226684,
0.67683853408712207040020226684, 1.42623141560295896873360950197, 2.57033956819625749359801695406, 4.78352572684092250172893484598, 5.31647800535557147901902226190, 6.30654655034413965750525685694, 7.43675540306470765851592682142, 7.68924676800510808601381565030, 8.817125646757025560550720334021, 9.649465342260983660693464750066