L(s) = 1 | − 3.20·2-s − 21.7·4-s − 26.7·5-s + 39.0·7-s + 172.·8-s + 85.6·10-s + 606.·11-s − 161.·13-s − 125.·14-s + 142.·16-s + 1.65e3·17-s + 882.·19-s + 580.·20-s − 1.94e3·22-s − 2.92e3·23-s − 2.41e3·25-s + 518.·26-s − 847.·28-s + 7.06e3·29-s − 2.25e3·31-s − 5.96e3·32-s − 5.29e3·34-s − 1.04e3·35-s − 7.56e3·37-s − 2.82e3·38-s − 4.60e3·40-s + 1.67e4·41-s + ⋯ |
L(s) = 1 | − 0.566·2-s − 0.678·4-s − 0.478·5-s + 0.301·7-s + 0.951·8-s + 0.270·10-s + 1.51·11-s − 0.265·13-s − 0.170·14-s + 0.139·16-s + 1.38·17-s + 0.560·19-s + 0.324·20-s − 0.856·22-s − 1.15·23-s − 0.771·25-s + 0.150·26-s − 0.204·28-s + 1.55·29-s − 0.421·31-s − 1.03·32-s − 0.785·34-s − 0.143·35-s − 0.908·37-s − 0.317·38-s − 0.454·40-s + 1.55·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.321750838\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.321750838\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
good | 2 | \( 1 + 3.20T + 32T^{2} \) |
| 5 | \( 1 + 26.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 39.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 606.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 161.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.65e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 882.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.92e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.25e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.67e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.50e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.40e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.10e4T + 4.18e8T^{2} \) |
| 61 | \( 1 - 3.47e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 26.9T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.31e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.28e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.66e4T + 8.58e9T^{2} \) |
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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
−9.793110422451868387782481867605, −9.303337803539264084120571349559, −8.176680914829408779756399545401, −7.73267376802423352967762319058, −6.52506849131434740032877624638, −5.31654307206667007447195059025, −4.25543155996953000858868847705, −3.47765007671054179789174802162, −1.62032381193267384516347852672, −0.67720617584632738513389113093,
0.67720617584632738513389113093, 1.62032381193267384516347852672, 3.47765007671054179789174802162, 4.25543155996953000858868847705, 5.31654307206667007447195059025, 6.52506849131434740032877624638, 7.73267376802423352967762319058, 8.176680914829408779756399545401, 9.303337803539264084120571349559, 9.793110422451868387782481867605