| L(s) = 1 | − 0.779·2-s − 3·3-s − 7.39·4-s + 18.3·5-s + 2.33·6-s + 12.7·7-s + 11.9·8-s + 9·9-s − 14.3·10-s + 11·11-s + 22.1·12-s − 73.8·13-s − 9.93·14-s − 55.1·15-s + 49.7·16-s − 2.09·17-s − 7.01·18-s − 84.5·19-s − 135.·20-s − 38.2·21-s − 8.57·22-s − 188.·23-s − 35.9·24-s + 212.·25-s + 57.5·26-s − 27·27-s − 94.2·28-s + ⋯ |
| L(s) = 1 | − 0.275·2-s − 0.577·3-s − 0.924·4-s + 1.64·5-s + 0.159·6-s + 0.688·7-s + 0.530·8-s + 0.333·9-s − 0.452·10-s + 0.301·11-s + 0.533·12-s − 1.57·13-s − 0.189·14-s − 0.949·15-s + 0.778·16-s − 0.0299·17-s − 0.0918·18-s − 1.02·19-s − 1.51·20-s − 0.397·21-s − 0.0830·22-s − 1.70·23-s − 0.306·24-s + 1.70·25-s + 0.434·26-s − 0.192·27-s − 0.636·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.588254248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.588254248\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 0.779T + 8T^{2} \) |
| 5 | \( 1 - 18.3T + 125T^{2} \) |
| 7 | \( 1 - 12.7T + 343T^{2} \) |
| 13 | \( 1 + 73.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.09T + 4.91e3T^{2} \) |
| 19 | \( 1 + 84.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 134.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 319.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 204.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 375.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 54.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 563.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 23.4T + 1.48e5T^{2} \) |
| 59 | \( 1 - 662.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 341.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 634.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 374.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 972.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 80.2T + 7.04e5T^{2} \) |
| 97 | \( 1 - 506.T + 9.12e5T^{2} \) |
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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
−8.881908123151900962366663276402, −8.190121320337590913270601541058, −7.21319180774636890180097238064, −6.21159945923804181924037499671, −5.63478728962457850225266788866, −4.74865600854932058274226135508, −4.30972233058560784836176394865, −2.52331144029346162369032317324, −1.74643594712865866123754822516, −0.63809470797070686633230391541,
0.63809470797070686633230391541, 1.74643594712865866123754822516, 2.52331144029346162369032317324, 4.30972233058560784836176394865, 4.74865600854932058274226135508, 5.63478728962457850225266788866, 6.21159945923804181924037499671, 7.21319180774636890180097238064, 8.190121320337590913270601541058, 8.881908123151900962366663276402
