L(s) = 1 | − 4.92·2-s − 3·3-s + 16.2·4-s + 10.2·5-s + 14.7·6-s + 17.4·7-s − 40.7·8-s + 9·9-s − 50.5·10-s + 11·11-s − 48.8·12-s + 72.1·13-s − 86.0·14-s − 30.7·15-s + 70.6·16-s − 57.0·17-s − 44.3·18-s + 4.76·19-s + 166.·20-s − 52.3·21-s − 54.1·22-s − 122.·23-s + 122.·24-s − 19.9·25-s − 355.·26-s − 27·27-s + 284.·28-s + ⋯ |
L(s) = 1 | − 1.74·2-s − 0.577·3-s + 2.03·4-s + 0.916·5-s + 1.00·6-s + 0.942·7-s − 1.80·8-s + 0.333·9-s − 1.59·10-s + 0.301·11-s − 1.17·12-s + 1.53·13-s − 1.64·14-s − 0.529·15-s + 1.10·16-s − 0.813·17-s − 0.580·18-s + 0.0575·19-s + 1.86·20-s − 0.544·21-s − 0.525·22-s − 1.11·23-s + 1.04·24-s − 0.159·25-s − 2.68·26-s − 0.192·27-s + 1.91·28-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
\(L(2)\) |
\(\approx\) |
\(1.060342251\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.060342251\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + 4.92T + 8T^{2} \) |
| 5 | \( 1 - 10.2T + 125T^{2} \) |
| 7 | \( 1 - 17.4T + 343T^{2} \) |
| 13 | \( 1 - 72.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 4.76T + 6.85e3T^{2} \) |
| 23 | \( 1 + 122.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 148.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 154.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 129.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 394.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 18.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 367.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 180.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 833.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 411.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 383.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 671.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 729.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.38e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.75e3T + 9.12e5T^{2} \) |
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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
−8.691214620358825221924343410229, −8.331184963360185217174732901848, −7.41092773885450091658475540539, −6.44157651383689009812940724527, −6.06444793462037309931561908724, −4.99009928187275964309000333456, −3.74400714086241489918873487492, −2.06999683417935053953464610950, −1.66078672485626543213202976681, −0.66200264823025723591960171986,
0.66200264823025723591960171986, 1.66078672485626543213202976681, 2.06999683417935053953464610950, 3.74400714086241489918873487492, 4.99009928187275964309000333456, 6.06444793462037309931561908724, 6.44157651383689009812940724527, 7.41092773885450091658475540539, 8.331184963360185217174732901848, 8.691214620358825221924343410229