L(s) = 1 | + 1.98·2-s + 2.42·3-s + 1.93·4-s − 0.826·5-s + 4.80·6-s + 0.383·7-s − 0.136·8-s + 2.86·9-s − 1.63·10-s + 3.48·11-s + 4.67·12-s − 0.283·13-s + 0.760·14-s − 2.00·15-s − 4.13·16-s − 4.93·17-s + 5.68·18-s − 1.37·19-s − 1.59·20-s + 0.929·21-s + 6.91·22-s + 4.28·23-s − 0.331·24-s − 4.31·25-s − 0.561·26-s − 0.317·27-s + 0.740·28-s + ⋯ |
L(s) = 1 | + 1.40·2-s + 1.39·3-s + 0.965·4-s − 0.369·5-s + 1.96·6-s + 0.144·7-s − 0.0483·8-s + 0.956·9-s − 0.518·10-s + 1.05·11-s + 1.35·12-s − 0.0785·13-s + 0.203·14-s − 0.516·15-s − 1.03·16-s − 1.19·17-s + 1.34·18-s − 0.314·19-s − 0.356·20-s + 0.202·21-s + 1.47·22-s + 0.892·23-s − 0.0676·24-s − 0.863·25-s − 0.110·26-s − 0.0610·27-s + 0.139·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.071807269\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.071807269\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 547 | \( 1 - T \) |
good | 2 | \( 1 - 1.98T + 2T^{2} \) |
| 3 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 + 0.826T + 5T^{2} \) |
| 7 | \( 1 - 0.383T + 7T^{2} \) |
| 11 | \( 1 - 3.48T + 11T^{2} \) |
| 13 | \( 1 + 0.283T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 + 1.37T + 19T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 2.57T + 29T^{2} \) |
| 31 | \( 1 - 8.23T + 31T^{2} \) |
| 37 | \( 1 - 1.31T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 + 2.33T + 43T^{2} \) |
| 47 | \( 1 + 3.00T + 47T^{2} \) |
| 53 | \( 1 + 6.14T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 3.13T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 + 3.73T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 + 2.61T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 - 2.11T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−11.22141317409874644498095098736, −9.700049726159845866469309897206, −8.932851140487065985312077515794, −8.196974513115901900266989635665, −7.02559640833648567315210605294, −6.20451317394968631824854064001, −4.75636201247294252293608887270, −4.01308399222230810345890542348, −3.18984908893420473839407497653, −2.10975171258152058186983214009,
2.10975171258152058186983214009, 3.18984908893420473839407497653, 4.01308399222230810345890542348, 4.75636201247294252293608887270, 6.20451317394968631824854064001, 7.02559640833648567315210605294, 8.196974513115901900266989635665, 8.932851140487065985312077515794, 9.700049726159845866469309897206, 11.22141317409874644498095098736