L(s) = 1 | + 1.61·2-s + 0.381·3-s + 0.618·4-s + 5-s + 0.618·6-s + 7-s − 2.23·8-s − 2.85·9-s + 1.61·10-s + 0.618·11-s + 0.236·12-s − 13-s + 1.61·14-s + 0.381·15-s − 4.85·16-s + 3.85·17-s − 4.61·18-s − 19-s + 0.618·20-s + 0.381·21-s + 1.00·22-s − 5.47·23-s − 0.854·24-s − 4·25-s − 1.61·26-s − 2.23·27-s + 0.618·28-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.220·3-s + 0.309·4-s + 0.447·5-s + 0.252·6-s + 0.377·7-s − 0.790·8-s − 0.951·9-s + 0.511·10-s + 0.186·11-s + 0.0681·12-s − 0.277·13-s + 0.432·14-s + 0.0986·15-s − 1.21·16-s + 0.934·17-s − 1.08·18-s − 0.229·19-s + 0.138·20-s + 0.0833·21-s + 0.213·22-s − 1.14·23-s − 0.174·24-s − 0.800·25-s − 0.317·26-s − 0.430·27-s + 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.818885651\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.818885651\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 0.618T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 23 | \( 1 + 5.47T + 23T^{2} \) |
| 29 | \( 1 - 1.38T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 8.47T + 43T^{2} \) |
| 47 | \( 1 - 7.47T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 - 7.76T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 - 8.70T + 71T^{2} \) |
| 73 | \( 1 + 11.3T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 3.14T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 - 7.47T + 97T^{2} \) |
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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
−13.62025060895511532214578037311, −12.25370563588666584498674988050, −11.71265111051365992000621233065, −10.22529099332162626706503366805, −9.053331509949530621405899261300, −7.936094927570823561051604323227, −6.20826058264757330489467877805, −5.40571077549911655311343714364, −4.05900994656425477202054392423, −2.62047472811705218653323646518,
2.62047472811705218653323646518, 4.05900994656425477202054392423, 5.40571077549911655311343714364, 6.20826058264757330489467877805, 7.936094927570823561051604323227, 9.053331509949530621405899261300, 10.22529099332162626706503366805, 11.71265111051365992000621233065, 12.25370563588666584498674988050, 13.62025060895511532214578037311