| L(s) = 1 | + 2.09·2-s + 3·3-s − 3.62·4-s + 15.8·5-s + 6.27·6-s − 15.4·7-s − 24.3·8-s + 9·9-s + 33.1·10-s + 11·11-s − 10.8·12-s + 68.1·13-s − 32.2·14-s + 47.5·15-s − 21.7·16-s − 16.4·17-s + 18.8·18-s + 67.0·19-s − 57.5·20-s − 46.2·21-s + 22.9·22-s + 140.·23-s − 72.9·24-s + 126.·25-s + 142.·26-s + 27·27-s + 55.9·28-s + ⋯ |
| L(s) = 1 | + 0.739·2-s + 0.577·3-s − 0.453·4-s + 1.41·5-s + 0.426·6-s − 0.832·7-s − 1.07·8-s + 0.333·9-s + 1.04·10-s + 0.301·11-s − 0.261·12-s + 1.45·13-s − 0.615·14-s + 0.819·15-s − 0.340·16-s − 0.234·17-s + 0.246·18-s + 0.809·19-s − 0.643·20-s − 0.480·21-s + 0.222·22-s + 1.27·23-s − 0.620·24-s + 1.01·25-s + 1.07·26-s + 0.192·27-s + 0.377·28-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
| \(L(2)\) |
\(\approx\) |
\(4.645499253\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.645499253\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 2.09T + 8T^{2} \) |
| 5 | \( 1 - 15.8T + 125T^{2} \) |
| 7 | \( 1 + 15.4T + 343T^{2} \) |
| 13 | \( 1 - 68.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 67.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 129.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 74.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 225.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 428.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 262.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 442.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 341.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 456.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 314.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 688.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.23e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 536.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 928.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 74.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 98.2T + 9.12e5T^{2} \) |
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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
−8.843746306148662565838988052799, −8.367783550394101183438405179934, −6.71964328053219540571745697675, −6.48874466075727863644247320439, −5.46980187031813253305922744353, −4.90884115298108535803242863159, −3.48700242777835854271865541046, −3.30578028263026325234154219756, −2.00483816322264080391719480787, −0.895218333182605992630130505613,
0.895218333182605992630130505613, 2.00483816322264080391719480787, 3.30578028263026325234154219756, 3.48700242777835854271865541046, 4.90884115298108535803242863159, 5.46980187031813253305922744353, 6.48874466075727863644247320439, 6.71964328053219540571745697675, 8.367783550394101183438405179934, 8.843746306148662565838988052799
