| L(s) = 1 | + 3.38·2-s + 9·3-s − 20.5·4-s − 54.5·5-s + 30.4·6-s − 177.·8-s + 81·9-s − 184.·10-s + 481.·11-s − 185.·12-s + 512.·13-s − 490.·15-s + 57.6·16-s + 590.·17-s + 273.·18-s + 2.45e3·19-s + 1.12e3·20-s + 1.62e3·22-s + 1.77e3·23-s − 1.59e3·24-s − 151.·25-s + 1.73e3·26-s + 729·27-s − 4.24e3·29-s − 1.65e3·30-s + 9.76e3·31-s + 5.88e3·32-s + ⋯ |
| L(s) = 1 | + 0.597·2-s + 0.577·3-s − 0.642·4-s − 0.975·5-s + 0.345·6-s − 0.981·8-s + 0.333·9-s − 0.582·10-s + 1.20·11-s − 0.371·12-s + 0.841·13-s − 0.563·15-s + 0.0562·16-s + 0.495·17-s + 0.199·18-s + 1.55·19-s + 0.627·20-s + 0.717·22-s + 0.699·23-s − 0.566·24-s − 0.0486·25-s + 0.502·26-s + 0.192·27-s − 0.937·29-s − 0.336·30-s + 1.82·31-s + 1.01·32-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
| \(L(3)\) |
\(\approx\) |
\(2.410768047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.410768047\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.38T + 32T^{2} \) |
| 5 | \( 1 + 54.5T + 3.12e3T^{2} \) |
| 11 | \( 1 - 481.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 512.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 590.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.45e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.77e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.24e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.96e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.32e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.48e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.15e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.28e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.59e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.71e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.03e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.58e9T^{2} \) |
| 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
−12.08506712238739041077955065231, −11.58576227581255801796436526470, −9.900755930783477652776684201677, −8.920282885554863131679088000891, −8.079173363749782300343284697197, −6.78023670760780428431404559975, −5.26870044376697009360016665070, −3.92712131000676326587413950302, −3.35130811350176570559939482912, −0.979127880825138800939830286034,
0.979127880825138800939830286034, 3.35130811350176570559939482912, 3.92712131000676326587413950302, 5.26870044376697009360016665070, 6.78023670760780428431404559975, 8.079173363749782300343284697197, 8.920282885554863131679088000891, 9.900755930783477652776684201677, 11.58576227581255801796436526470, 12.08506712238739041077955065231
