L(s) = 1 | + 9·3-s − 77.6·5-s + 49·7-s + 81·9-s − 477.·11-s − 63.7·13-s − 698.·15-s + 1.03e3·17-s + 667.·19-s + 441·21-s − 3.25e3·23-s + 2.90e3·25-s + 729·27-s + 2.30e3·29-s − 3.71e3·31-s − 4.29e3·33-s − 3.80e3·35-s + 1.22e4·37-s − 573.·39-s − 1.82e3·41-s + 2.07e4·43-s − 6.28e3·45-s + 4.28e3·47-s + 2.40e3·49-s + 9.33e3·51-s + 2.57e4·53-s + 3.70e4·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.38·5-s + 0.377·7-s + 0.333·9-s − 1.18·11-s − 0.104·13-s − 0.801·15-s + 0.870·17-s + 0.423·19-s + 0.218·21-s − 1.28·23-s + 0.928·25-s + 0.192·27-s + 0.508·29-s − 0.694·31-s − 0.686·33-s − 0.524·35-s + 1.47·37-s − 0.0604·39-s − 0.169·41-s + 1.71·43-s − 0.462·45-s + 0.282·47-s + 0.142·49-s + 0.502·51-s + 1.25·53-s + 1.65·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.698325318\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.698325318\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 9T \) |
| 7 | \( 1 - 49T \) |
good | 5 | \( 1 + 77.6T + 3.12e3T^{2} \) |
| 11 | \( 1 + 477.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 63.7T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.03e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 667.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.30e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.71e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.07e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 4.28e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.83e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.68e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.25e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.23e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.56e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.60e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.35e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.42e5T + 8.58e9T^{2} \) |
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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
−10.73788599517428886758987050178, −9.838091991585157595872018825899, −8.579527667719131918440022482136, −7.77020517252972137557275260862, −7.44018738515136511795267339551, −5.72192341657863177969015414653, −4.50468082853022188551645698945, −3.57446630670126067340443351805, −2.42105519681349864822015371746, −0.68576481482159037703123855094,
0.68576481482159037703123855094, 2.42105519681349864822015371746, 3.57446630670126067340443351805, 4.50468082853022188551645698945, 5.72192341657863177969015414653, 7.44018738515136511795267339551, 7.77020517252972137557275260862, 8.579527667719131918440022482136, 9.838091991585157595872018825899, 10.73788599517428886758987050178