| L(s) = 1 | + 1.90·2-s + 0.688·3-s + 1.62·4-s + 0.688·5-s + 1.31·6-s − 7-s − 0.719·8-s − 2.52·9-s + 1.31·10-s − 0.903·11-s + 1.11·12-s + 0.622·13-s − 1.90·14-s + 0.474·15-s − 4.61·16-s + 1.31·17-s − 4.80·18-s + 7.05·19-s + 1.11·20-s − 0.688·21-s − 1.71·22-s + 23-s − 0.495·24-s − 4.52·25-s + 1.18·26-s − 3.80·27-s − 1.62·28-s + ⋯ |
| L(s) = 1 | + 1.34·2-s + 0.397·3-s + 0.811·4-s + 0.308·5-s + 0.535·6-s − 0.377·7-s − 0.254·8-s − 0.841·9-s + 0.414·10-s − 0.272·11-s + 0.322·12-s + 0.172·13-s − 0.508·14-s + 0.122·15-s − 1.15·16-s + 0.317·17-s − 1.13·18-s + 1.61·19-s + 0.249·20-s − 0.150·21-s − 0.366·22-s + 0.208·23-s − 0.101·24-s − 0.905·25-s + 0.232·26-s − 0.732·27-s − 0.306·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.169804218\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.169804218\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.90T + 2T^{2} \) |
| 3 | \( 1 - 0.688T + 3T^{2} \) |
| 5 | \( 1 - 0.688T + 5T^{2} \) |
| 11 | \( 1 + 0.903T + 11T^{2} \) |
| 13 | \( 1 - 0.622T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 - 7.05T + 19T^{2} \) |
| 29 | \( 1 - 1.52T + 29T^{2} \) |
| 31 | \( 1 + 3.54T + 31T^{2} \) |
| 37 | \( 1 - 2.42T + 37T^{2} \) |
| 41 | \( 1 + 4.75T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 9.73T + 47T^{2} \) |
| 53 | \( 1 + 6.85T + 53T^{2} \) |
| 59 | \( 1 - 9.97T + 59T^{2} \) |
| 61 | \( 1 - 7.87T + 61T^{2} \) |
| 67 | \( 1 + 5.76T + 67T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + 1.86T + 73T^{2} \) |
| 79 | \( 1 - 7.65T + 79T^{2} \) |
| 83 | \( 1 + 11.0T + 83T^{2} \) |
| 89 | \( 1 + 2.98T + 89T^{2} \) |
| 97 | \( 1 + 2.49T + 97T^{2} \) |
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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
−13.14446776459336635904302526611, −12.09907248256119464652281466351, −11.28770868384557649052007499889, −9.812237284427036848244126087118, −8.839254133771052909051243973350, −7.44256206948148874183163941005, −6.00199631734413905374414197200, −5.28492078504906890939946069597, −3.71715355489902784426571498999, −2.70320216301554500345995445588,
2.70320216301554500345995445588, 3.71715355489902784426571498999, 5.28492078504906890939946069597, 6.00199631734413905374414197200, 7.44256206948148874183163941005, 8.839254133771052909051243973350, 9.812237284427036848244126087118, 11.28770868384557649052007499889, 12.09907248256119464652281466351, 13.14446776459336635904302526611
