L(s) = 1 | − 0.0402·2-s + 9·3-s − 31.9·4-s + 25·5-s − 0.362·6-s + 37.9·7-s + 2.57·8-s + 81·9-s − 1.00·10-s + 121·11-s − 287.·12-s − 338.·13-s − 1.52·14-s + 225·15-s + 1.02e3·16-s + 57.1·17-s − 3.26·18-s − 1.25e3·19-s − 799.·20-s + 341.·21-s − 4.87·22-s + 3.45e3·23-s + 23.1·24-s + 625·25-s + 13.6·26-s + 729·27-s − 1.21e3·28-s + ⋯ |
L(s) = 1 | − 0.00711·2-s + 0.577·3-s − 0.999·4-s + 0.447·5-s − 0.00410·6-s + 0.292·7-s + 0.0142·8-s + 0.333·9-s − 0.00318·10-s + 0.301·11-s − 0.577·12-s − 0.556·13-s − 0.00208·14-s + 0.258·15-s + 0.999·16-s + 0.0479·17-s − 0.00237·18-s − 0.795·19-s − 0.447·20-s + 0.168·21-s − 0.00214·22-s + 1.36·23-s + 0.00821·24-s + 0.200·25-s + 0.00395·26-s + 0.192·27-s − 0.292·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.148460085\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.148460085\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 9T \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 0.0402T + 32T^{2} \) |
| 7 | \( 1 - 37.9T + 1.68e4T^{2} \) |
| 13 | \( 1 + 338.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 57.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.25e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.48e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.77e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.95e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 191.T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.78e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.17e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.74e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.43e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.50e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.76e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.74e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.01e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−12.19165817421432728885374092873, −10.68367173445096610368283111839, −9.722953131886187081998509253140, −8.888793912678377811063528483907, −8.053726988591102479194941650520, −6.67302117329865505942644966096, −5.17146295175415660177877482287, −4.19573360457166363137367963801, −2.67514575040714572762460444432, −0.976214548276323367048440392879,
0.976214548276323367048440392879, 2.67514575040714572762460444432, 4.19573360457166363137367963801, 5.17146295175415660177877482287, 6.67302117329865505942644966096, 8.053726988591102479194941650520, 8.888793912678377811063528483907, 9.722953131886187081998509253140, 10.68367173445096610368283111839, 12.19165817421432728885374092873