| L(s) = 1 | − 3.44·2-s − 3·3-s + 3.86·4-s − 21.1·5-s + 10.3·6-s − 30.8·7-s + 14.2·8-s + 9·9-s + 73.0·10-s + 11·11-s − 11.5·12-s − 6.64·13-s + 106.·14-s + 63.5·15-s − 79.9·16-s + 112.·17-s − 30.9·18-s + 128.·19-s − 81.8·20-s + 92.5·21-s − 37.8·22-s − 103.·23-s − 42.7·24-s + 324.·25-s + 22.8·26-s − 27·27-s − 119.·28-s + ⋯ |
| L(s) = 1 | − 1.21·2-s − 0.577·3-s + 0.482·4-s − 1.89·5-s + 0.703·6-s − 1.66·7-s + 0.629·8-s + 0.333·9-s + 2.30·10-s + 0.301·11-s − 0.278·12-s − 0.141·13-s + 2.02·14-s + 1.09·15-s − 1.24·16-s + 1.60·17-s − 0.405·18-s + 1.54·19-s − 0.915·20-s + 0.961·21-s − 0.367·22-s − 0.937·23-s − 0.363·24-s + 2.59·25-s + 0.172·26-s − 0.192·27-s − 0.804·28-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
| \(L(2)\) |
\(\approx\) |
\(0.2864651606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2864651606\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 3.44T + 8T^{2} \) |
| 5 | \( 1 + 21.1T + 125T^{2} \) |
| 7 | \( 1 + 30.8T + 343T^{2} \) |
| 13 | \( 1 + 6.64T + 2.19e3T^{2} \) |
| 17 | \( 1 - 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 128.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 103.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 67.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 75.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 116.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 169.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 59.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 247.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 14.2T + 2.05e5T^{2} \) |
| 67 | \( 1 + 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 353.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 861.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 901.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 937.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 434.T + 9.12e5T^{2} \) |
| 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
−8.792220329540255698798621610479, −7.905221239176649705200321216671, −7.44615531906609120643958010791, −6.83042387621108003470859643661, −5.79181113680501807514956047947, −4.62680353941983652012340441786, −3.68295975910077890176953872379, −3.09928917105961081649721042078, −1.09267521387806396539868068212, −0.38512876364971877118003716841,
0.38512876364971877118003716841, 1.09267521387806396539868068212, 3.09928917105961081649721042078, 3.68295975910077890176953872379, 4.62680353941983652012340441786, 5.79181113680501807514956047947, 6.83042387621108003470859643661, 7.44615531906609120643958010791, 7.905221239176649705200321216671, 8.792220329540255698798621610479
