L(s) = 1 | + 10.0·2-s + 9·3-s + 68.1·4-s + 70.3·5-s + 90.0·6-s + 362.·8-s + 81·9-s + 704.·10-s − 731.·11-s + 613.·12-s + 899.·13-s + 633.·15-s + 1.44e3·16-s − 1.39e3·17-s + 810.·18-s − 190.·19-s + 4.79e3·20-s − 7.32e3·22-s + 42.9·23-s + 3.25e3·24-s + 1.82e3·25-s + 9.00e3·26-s + 729·27-s − 7.74e3·29-s + 6.33e3·30-s + 1.17e3·31-s + 2.85e3·32-s + ⋯ |
L(s) = 1 | + 1.76·2-s + 0.577·3-s + 2.13·4-s + 1.25·5-s + 1.02·6-s + 2.00·8-s + 0.333·9-s + 2.22·10-s − 1.82·11-s + 1.23·12-s + 1.47·13-s + 0.726·15-s + 1.40·16-s − 1.16·17-s + 0.589·18-s − 0.120·19-s + 2.68·20-s − 3.22·22-s + 0.0169·23-s + 1.15·24-s + 0.583·25-s + 2.61·26-s + 0.192·27-s − 1.71·29-s + 1.28·30-s + 0.220·31-s + 0.492·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(7.593944550\) |
\(L(\frac12)\) |
\(\approx\) |
\(7.593944550\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 10.0T + 32T^{2} \) |
| 5 | \( 1 - 70.3T + 3.12e3T^{2} \) |
| 11 | \( 1 + 731.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 899.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.39e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 190.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 42.9T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.74e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.17e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.28e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.03e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 3.24e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.64e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.43e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.46e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.72e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.99e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.68e4T + 8.58e9T^{2} \) |
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
−12.89788777813784850596785384575, −11.20325758307028494722933586657, −10.48696662289040963713905720170, −9.058336243079591994006273234150, −7.59795075351536710752835566600, −6.19664887741897577890375479623, −5.53535069728199807220933163874, −4.25407851635170220253298790363, −2.84420823570864719464230548146, −1.97136005584661824083699038581,
1.97136005584661824083699038581, 2.84420823570864719464230548146, 4.25407851635170220253298790363, 5.53535069728199807220933163874, 6.19664887741897577890375479623, 7.59795075351536710752835566600, 9.058336243079591994006273234150, 10.48696662289040963713905720170, 11.20325758307028494722933586657, 12.89788777813784850596785384575