| L(s) = 1 | + 2.66·2-s − 1.22·3-s + 5.11·4-s − 1.15·5-s − 3.25·6-s + 2.36·7-s + 8.30·8-s − 1.50·9-s − 3.08·10-s + 6.51·11-s − 6.24·12-s − 3.92·13-s + 6.32·14-s + 1.41·15-s + 11.9·16-s + 17-s − 4.02·18-s + 0.308·19-s − 5.92·20-s − 2.89·21-s + 17.3·22-s + 1.28·23-s − 10.1·24-s − 3.65·25-s − 10.4·26-s + 5.50·27-s + 12.1·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.705·3-s + 2.55·4-s − 0.517·5-s − 1.32·6-s + 0.895·7-s + 2.93·8-s − 0.502·9-s − 0.976·10-s + 1.96·11-s − 1.80·12-s − 1.08·13-s + 1.68·14-s + 0.365·15-s + 2.98·16-s + 0.242·17-s − 0.948·18-s + 0.0706·19-s − 1.32·20-s − 0.631·21-s + 3.70·22-s + 0.268·23-s − 2.07·24-s − 0.731·25-s − 2.05·26-s + 1.05·27-s + 2.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.875936782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.875936782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 - 2.66T + 2T^{2} \) |
| 3 | \( 1 + 1.22T + 3T^{2} \) |
| 5 | \( 1 + 1.15T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 - 6.51T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 19 | \( 1 - 0.308T + 19T^{2} \) |
| 23 | \( 1 - 1.28T + 23T^{2} \) |
| 29 | \( 1 + 2.29T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 + 5.52T + 37T^{2} \) |
| 41 | \( 1 + 0.358T + 41T^{2} \) |
| 47 | \( 1 + 7.52T + 47T^{2} \) |
| 53 | \( 1 + 0.0422T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 + 4.20T + 61T^{2} \) |
| 67 | \( 1 - 7.62T + 67T^{2} \) |
| 71 | \( 1 - 6.07T + 71T^{2} \) |
| 73 | \( 1 + 7.32T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 7.59T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−11.14423750927714447890649077535, −9.838367498521508765145834279614, −8.405027173879089943826659701591, −7.29842951581079891265907489716, −6.57301051678800467669373893147, −5.71374013423581771093794939434, −4.85010858350859939805606671040, −4.19337375422655648897726780367, −3.14107822008402503345805727401, −1.66201749664512394124435008946,
1.66201749664512394124435008946, 3.14107822008402503345805727401, 4.19337375422655648897726780367, 4.85010858350859939805606671040, 5.71374013423581771093794939434, 6.57301051678800467669373893147, 7.29842951581079891265907489716, 8.405027173879089943826659701591, 9.838367498521508765145834279614, 11.14423750927714447890649077535
