L(s) = 1 | + 1.70·2-s − 3·3-s − 5.10·4-s + 17.1·5-s − 5.10·6-s − 24.4·7-s − 22.2·8-s + 9·9-s + 29.2·10-s − 11·11-s + 15.3·12-s − 72.7·13-s − 41.5·14-s − 51.5·15-s + 2.92·16-s − 61.9·17-s + 15.3·18-s − 26.2·19-s − 87.6·20-s + 73.2·21-s − 18.7·22-s − 177.·23-s + 66.8·24-s + 169.·25-s − 123.·26-s − 27·27-s + 124.·28-s + ⋯ |
L(s) = 1 | + 0.601·2-s − 0.577·3-s − 0.638·4-s + 1.53·5-s − 0.347·6-s − 1.31·7-s − 0.985·8-s + 0.333·9-s + 0.923·10-s − 0.301·11-s + 0.368·12-s − 1.55·13-s − 0.792·14-s − 0.886·15-s + 0.0457·16-s − 0.883·17-s + 0.200·18-s − 0.317·19-s − 0.980·20-s + 0.761·21-s − 0.181·22-s − 1.61·23-s + 0.568·24-s + 1.35·25-s − 0.932·26-s − 0.192·27-s + 0.841·28-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
\(L(2)\) |
\(\approx\) |
\(0.8347824596\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8347824596\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.70T + 8T^{2} \) |
| 5 | \( 1 - 17.1T + 125T^{2} \) |
| 7 | \( 1 + 24.4T + 343T^{2} \) |
| 13 | \( 1 + 72.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 61.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 177.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 143.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 390.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 23.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 128.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 130.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 362.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 418.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 360.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 877.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 313.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 933.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 87.3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 106.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 738.T + 9.12e5T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.257744557111007217451566136298, −7.982464409675132583882309480114, −6.83011171033371265680568071811, −6.10905825190533102411819006707, −5.74304859297392630831292001263, −4.84738084258034340028230706438, −4.07464189701973621289202647950, −2.81131162609535556183275775289, −2.09952629073073132951031980882, −0.36981876021336517993632557321,
0.36981876021336517993632557321, 2.09952629073073132951031980882, 2.81131162609535556183275775289, 4.07464189701973621289202647950, 4.84738084258034340028230706438, 5.74304859297392630831292001263, 6.10905825190533102411819006707, 6.83011171033371265680568071811, 7.982464409675132583882309480114, 9.257744557111007217451566136298