L(s) = 1 | + 0.898·2-s + 9·3-s − 31.1·4-s + 8.08·6-s − 120.·7-s − 56.7·8-s + 81·9-s − 121·11-s − 280.·12-s − 667.·13-s − 108.·14-s + 947.·16-s + 74.5·17-s + 72.7·18-s − 708.·19-s − 1.08e3·21-s − 108.·22-s − 1.23e3·23-s − 510.·24-s − 599.·26-s + 729·27-s + 3.76e3·28-s − 3.85e3·29-s − 8.23e3·31-s + 2.66e3·32-s − 1.08e3·33-s + 66.9·34-s + ⋯ |
L(s) = 1 | + 0.158·2-s + 0.577·3-s − 0.974·4-s + 0.0916·6-s − 0.931·7-s − 0.313·8-s + 0.333·9-s − 0.301·11-s − 0.562·12-s − 1.09·13-s − 0.147·14-s + 0.925·16-s + 0.0625·17-s + 0.0529·18-s − 0.450·19-s − 0.537·21-s − 0.0478·22-s − 0.488·23-s − 0.181·24-s − 0.173·26-s + 0.192·27-s + 0.907·28-s − 0.851·29-s − 1.53·31-s + 0.460·32-s − 0.174·33-s + 0.00993·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8864902512\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8864902512\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 0.898T + 32T^{2} \) |
| 7 | \( 1 + 120.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 667.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 74.5T + 1.41e6T^{2} \) |
| 19 | \( 1 + 708.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.23e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.23e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 672.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.60e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 7.57e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.42e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.31e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.80e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.24e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.23e3T + 2.07e9T^{2} \) |
| 79 | \( 1 - 4.70e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.44e4T + 8.58e9T^{2} \) |
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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
−9.635169177109589762092093514964, −8.721818076271486992914218239904, −7.86390861206893709739217110622, −7.00434839510848295871340781721, −5.86113685908143127961883935459, −4.95299266710530599508590960238, −3.95053997147718683355664420903, −3.17423126949604996795780205144, −2.03652086600441643384529890690, −0.38811719153106710876042454700,
0.38811719153106710876042454700, 2.03652086600441643384529890690, 3.17423126949604996795780205144, 3.95053997147718683355664420903, 4.95299266710530599508590960238, 5.86113685908143127961883935459, 7.00434839510848295871340781721, 7.86390861206893709739217110622, 8.721818076271486992914218239904, 9.635169177109589762092093514964