| L(s) = 1 | + 3.59·2-s − 1.67·3-s + 4.91·4-s − 9.72·5-s − 6.02·6-s − 15.7·7-s − 11.0·8-s − 24.1·9-s − 34.9·10-s − 59.3·11-s − 8.24·12-s + 13.2·13-s − 56.6·14-s + 16.3·15-s − 79.1·16-s − 29.0·17-s − 86.9·18-s − 103.·19-s − 47.7·20-s + 26.4·21-s − 213.·22-s + 8.54·23-s + 18.6·24-s − 30.4·25-s + 47.7·26-s + 85.8·27-s − 77.4·28-s + ⋯ |
| L(s) = 1 | + 1.27·2-s − 0.322·3-s + 0.614·4-s − 0.869·5-s − 0.410·6-s − 0.850·7-s − 0.490·8-s − 0.895·9-s − 1.10·10-s − 1.62·11-s − 0.198·12-s + 0.283·13-s − 1.08·14-s + 0.280·15-s − 1.23·16-s − 0.414·17-s − 1.13·18-s − 1.25·19-s − 0.534·20-s + 0.274·21-s − 2.06·22-s + 0.0774·23-s + 0.158·24-s − 0.243·25-s + 0.359·26-s + 0.612·27-s − 0.522·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1661492227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1661492227\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 3.59T + 8T^{2} \) |
| 3 | \( 1 + 1.67T + 27T^{2} \) |
| 5 | \( 1 + 9.72T + 125T^{2} \) |
| 7 | \( 1 + 15.7T + 343T^{2} \) |
| 11 | \( 1 + 59.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 13.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 29.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 103.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 8.54T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 328.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 47 | \( 1 + 161.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 171.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 789.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 520.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 444.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 856.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 969.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 986.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 121.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 691.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 503.T + 9.12e5T^{2} \) |
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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
−8.629941663315534777321134361602, −8.170661978169013475165330464359, −7.01237053005183703640834616184, −6.24435055055058594821986749329, −5.57734184743767687803488081125, −4.78831631982750864115705456151, −3.97365624748760591231842527196, −3.09743442491753703706700194016, −2.46206471649445698373695748842, −0.14538442847082076064107796853,
0.14538442847082076064107796853, 2.46206471649445698373695748842, 3.09743442491753703706700194016, 3.97365624748760591231842527196, 4.78831631982750864115705456151, 5.57734184743767687803488081125, 6.24435055055058594821986749329, 7.01237053005183703640834616184, 8.170661978169013475165330464359, 8.629941663315534777321134361602
