L(s) = 1 | − 2.19·2-s + 0.245·3-s + 2.83·4-s + 2.71·5-s − 0.540·6-s − 2.14·7-s − 1.82·8-s − 2.93·9-s − 5.96·10-s + 2.88·11-s + 0.695·12-s + 6.09·13-s + 4.71·14-s + 0.667·15-s − 1.64·16-s + 6.48·17-s + 6.46·18-s − 1.94·19-s + 7.68·20-s − 0.526·21-s − 6.34·22-s + 6.50·23-s − 0.448·24-s + 2.37·25-s − 13.3·26-s − 1.45·27-s − 6.06·28-s + ⋯ |
L(s) = 1 | − 1.55·2-s + 0.141·3-s + 1.41·4-s + 1.21·5-s − 0.220·6-s − 0.810·7-s − 0.645·8-s − 0.979·9-s − 1.88·10-s + 0.869·11-s + 0.200·12-s + 1.69·13-s + 1.25·14-s + 0.172·15-s − 0.412·16-s + 1.57·17-s + 1.52·18-s − 0.445·19-s + 1.71·20-s − 0.114·21-s − 1.35·22-s + 1.35·23-s − 0.0915·24-s + 0.475·25-s − 2.62·26-s − 0.280·27-s − 1.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6799700131\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6799700131\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 163 | \( 1 - T \) |
good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 0.245T + 3T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 2.14T + 7T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 + 1.94T + 19T^{2} \) |
| 23 | \( 1 - 6.50T + 23T^{2} \) |
| 29 | \( 1 + 4.59T + 29T^{2} \) |
| 31 | \( 1 + 3.00T + 31T^{2} \) |
| 37 | \( 1 - 0.110T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.881T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 1.74T + 53T^{2} \) |
| 59 | \( 1 + 10.8T + 59T^{2} \) |
| 61 | \( 1 + 8.64T + 61T^{2} \) |
| 67 | \( 1 - 3.01T + 67T^{2} \) |
| 71 | \( 1 + 0.761T + 71T^{2} \) |
| 73 | \( 1 + 3.27T + 73T^{2} \) |
| 79 | \( 1 - 2.69T + 79T^{2} \) |
| 83 | \( 1 + 0.451T + 83T^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−12.89556773828683658429798213668, −11.37608071812821010765392526566, −10.60279326383621919183001338454, −9.397152004002139557706003897973, −9.169914568906663880607335850759, −8.011667180712368257841976747115, −6.54162180381557249705375139075, −5.81156106012233173079506868661, −3.22806043850852759037456532922, −1.43247209006502686423966037383,
1.43247209006502686423966037383, 3.22806043850852759037456532922, 5.81156106012233173079506868661, 6.54162180381557249705375139075, 8.011667180712368257841976747115, 9.169914568906663880607335850759, 9.397152004002139557706003897973, 10.60279326383621919183001338454, 11.37608071812821010765392526566, 12.89556773828683658429798213668