| L(s) = 1 | + 9.37·2-s + 9·3-s + 55.8·4-s + 0.277·5-s + 84.3·6-s − 105.·7-s + 223.·8-s + 81·9-s + 2.59·10-s − 121·11-s + 502.·12-s + 147.·13-s − 984.·14-s + 2.49·15-s + 307.·16-s − 1.43e3·17-s + 759.·18-s + 2.03e3·19-s + 15.4·20-s − 945.·21-s − 1.13e3·22-s + 828.·23-s + 2.01e3·24-s − 3.12e3·25-s + 1.38e3·26-s + 729·27-s − 5.86e3·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.577·3-s + 1.74·4-s + 0.00495·5-s + 0.956·6-s − 0.810·7-s + 1.23·8-s + 0.333·9-s + 0.00821·10-s − 0.301·11-s + 1.00·12-s + 0.242·13-s − 1.34·14-s + 0.00286·15-s + 0.299·16-s − 1.20·17-s + 0.552·18-s + 1.29·19-s + 0.00865·20-s − 0.467·21-s − 0.499·22-s + 0.326·23-s + 0.712·24-s − 0.999·25-s + 0.401·26-s + 0.192·27-s − 1.41·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.869507390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.869507390\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 9T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 9.37T + 32T^{2} \) |
| 5 | \( 1 - 0.277T + 3.12e3T^{2} \) |
| 7 | \( 1 + 105.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 147.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.43e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 828.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.63e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.13e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.82e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.38e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.29e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.43e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.08e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.62e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.93e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.12e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.33e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.03e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.49e5T + 8.58e9T^{2} \) |
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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
−15.50465083808900918992923520188, −14.21788260677073977346304498900, −13.36877286391376741900515804397, −12.51282127289383014778852372457, −11.08628983909474475216419951988, −9.307421501695817151644839556853, −7.25040400667280707185793066363, −5.80663917185888314720906691634, −4.09020043256355535371557988808, −2.70176463886111205392393810501,
2.70176463886111205392393810501, 4.09020043256355535371557988808, 5.80663917185888314720906691634, 7.25040400667280707185793066363, 9.307421501695817151644839556853, 11.08628983909474475216419951988, 12.51282127289383014778852372457, 13.36877286391376741900515804397, 14.21788260677073977346304498900, 15.50465083808900918992923520188
