| L(s) = 1 | + 5.31·2-s + 20.2·4-s + 5·5-s − 13.4·7-s + 65.1·8-s + 26.5·10-s + 46.9·11-s − 36.1·13-s − 71.4·14-s + 184.·16-s + 54.6·17-s + 111.·19-s + 101.·20-s + 249.·22-s + 35.9·23-s + 25·25-s − 192.·26-s − 272.·28-s − 58.1·29-s − 295.·31-s + 458.·32-s + 290.·34-s − 67.1·35-s − 53.0·37-s + 591.·38-s + 325.·40-s − 128.·41-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.53·4-s + 0.447·5-s − 0.725·7-s + 2.87·8-s + 0.840·10-s + 1.28·11-s − 0.770·13-s − 1.36·14-s + 2.87·16-s + 0.779·17-s + 1.34·19-s + 1.13·20-s + 2.41·22-s + 0.326·23-s + 0.200·25-s − 1.44·26-s − 1.83·28-s − 0.372·29-s − 1.71·31-s + 2.53·32-s + 1.46·34-s − 0.324·35-s − 0.235·37-s + 2.52·38-s + 1.28·40-s − 0.488·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.686789664\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.686789664\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 - 5.31T + 8T^{2} \) |
| 7 | \( 1 + 13.4T + 343T^{2} \) |
| 11 | \( 1 - 46.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 36.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 111.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 35.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 58.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 295.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 53.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 128.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 87.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 479.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 635.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 48.0T + 2.26e5T^{2} \) |
| 67 | \( 1 + 28.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 576.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 835.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 203.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 464.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 993.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 881.T + 9.12e5T^{2} \) |
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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
−11.28760546133583254945866307823, −10.08856374234573974621487812824, −9.255161017167824126451993280697, −7.43595725733957509561741454221, −6.76999129326345828796136667438, −5.80471288431238686205647835172, −5.06044731752337269234862956210, −3.77669688988645145085316105998, −3.04068617154193777979897618394, −1.60773687503104671799500724727,
1.60773687503104671799500724727, 3.04068617154193777979897618394, 3.77669688988645145085316105998, 5.06044731752337269234862956210, 5.80471288431238686205647835172, 6.76999129326345828796136667438, 7.43595725733957509561741454221, 9.255161017167824126451993280697, 10.08856374234573974621487812824, 11.28760546133583254945866307823
