| L(s) = 1 | − 4.44·2-s + 3·3-s + 11.7·4-s − 6.52·5-s − 13.3·6-s − 19.5·7-s − 16.7·8-s + 9·9-s + 29.0·10-s + 11·11-s + 35.3·12-s + 12.3·13-s + 86.9·14-s − 19.5·15-s − 19.5·16-s − 84.6·17-s − 40.0·18-s + 26.7·19-s − 76.8·20-s − 58.6·21-s − 48.9·22-s − 81.4·23-s − 50.3·24-s − 82.4·25-s − 55.1·26-s + 27·27-s − 230.·28-s + ⋯ |
| L(s) = 1 | − 1.57·2-s + 0.577·3-s + 1.47·4-s − 0.583·5-s − 0.907·6-s − 1.05·7-s − 0.741·8-s + 0.333·9-s + 0.917·10-s + 0.301·11-s + 0.849·12-s + 0.264·13-s + 1.65·14-s − 0.336·15-s − 0.305·16-s − 1.20·17-s − 0.524·18-s + 0.323·19-s − 0.858·20-s − 0.609·21-s − 0.474·22-s − 0.738·23-s − 0.428·24-s − 0.659·25-s − 0.415·26-s + 0.192·27-s − 1.55·28-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
| \(L(2)\) |
\(\approx\) |
\(0.4680717953\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4680717953\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.44T + 8T^{2} \) |
| 5 | \( 1 + 6.52T + 125T^{2} \) |
| 7 | \( 1 + 19.5T + 343T^{2} \) |
| 13 | \( 1 - 12.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 84.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 81.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 171.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 106.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 165.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 407.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 2.83T + 7.95e4T^{2} \) |
| 47 | \( 1 + 105.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 347.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 729.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 176.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 751.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 290.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 957.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.04e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.44e3T + 9.12e5T^{2} \) |
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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
−8.722144050185555149683589208715, −8.304353981201203144231060452440, −7.40974955213610973094951290492, −6.81060350512100598008469502288, −6.08055990401801593548863167758, −4.50824793352755491921746215830, −3.62575664351084789157727922025, −2.63584791657164235243857903723, −1.62523762330906131206232136972, −0.38587968068364561432670785181,
0.38587968068364561432670785181, 1.62523762330906131206232136972, 2.63584791657164235243857903723, 3.62575664351084789157727922025, 4.50824793352755491921746215830, 6.08055990401801593548863167758, 6.81060350512100598008469502288, 7.40974955213610973094951290492, 8.304353981201203144231060452440, 8.722144050185555149683589208715
