| L(s) = 1 | + 1.16·2-s + 3·3-s − 6.64·4-s − 14.8·5-s + 3.49·6-s + 18.5·7-s − 17.0·8-s + 9·9-s − 17.3·10-s + 11·11-s − 19.9·12-s − 15.5·13-s + 21.6·14-s − 44.6·15-s + 33.3·16-s − 124.·17-s + 10.4·18-s − 6.25·19-s + 98.8·20-s + 55.6·21-s + 12.8·22-s + 107.·23-s − 51.1·24-s + 96.2·25-s − 18.1·26-s + 27·27-s − 123.·28-s + ⋯ |
| L(s) = 1 | + 0.411·2-s + 0.577·3-s − 0.830·4-s − 1.33·5-s + 0.237·6-s + 1.00·7-s − 0.753·8-s + 0.333·9-s − 0.547·10-s + 0.301·11-s − 0.479·12-s − 0.331·13-s + 0.412·14-s − 0.768·15-s + 0.520·16-s − 1.78·17-s + 0.137·18-s − 0.0755·19-s + 1.10·20-s + 0.578·21-s + 0.124·22-s + 0.971·23-s − 0.435·24-s + 0.770·25-s − 0.136·26-s + 0.192·27-s − 0.832·28-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
| \(L(2)\) |
\(\approx\) |
\(1.534618044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534618044\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.16T + 8T^{2} \) |
| 5 | \( 1 + 14.8T + 125T^{2} \) |
| 7 | \( 1 - 18.5T + 343T^{2} \) |
| 13 | \( 1 + 15.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.25T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 121.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 256.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 65.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 451.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 58.5T + 1.48e5T^{2} \) |
| 59 | \( 1 - 466.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 25.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 823.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 600.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.26e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.32e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−8.772120917713329631761033724386, −8.115604595843380830574822675562, −7.40938534113114704994098246434, −6.56830000347638828301359246457, −5.17439174671289163698297856154, −4.58751619735481608492543581955, −3.99254003899042569385228961418, −3.19168077288858748745872132477, −1.92807446152872083386727353416, −0.51602909333929794473209998045,
0.51602909333929794473209998045, 1.92807446152872083386727353416, 3.19168077288858748745872132477, 3.99254003899042569385228961418, 4.58751619735481608492543581955, 5.17439174671289163698297856154, 6.56830000347638828301359246457, 7.40938534113114704994098246434, 8.115604595843380830574822675562, 8.772120917713329631761033724386
