| L(s) = 1 | + 2-s − 0.905·3-s + 4-s − 1.11·5-s − 0.905·6-s − 1.83·7-s + 8-s − 2.17·9-s − 1.11·10-s + 3.15·11-s − 0.905·12-s − 6.12·13-s − 1.83·14-s + 1.00·15-s + 16-s + 1.33·17-s − 2.17·18-s + 1.55·19-s − 1.11·20-s + 1.65·21-s + 3.15·22-s + 5.24·23-s − 0.905·24-s − 3.76·25-s − 6.12·26-s + 4.69·27-s − 1.83·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.522·3-s + 0.5·4-s − 0.496·5-s − 0.369·6-s − 0.692·7-s + 0.353·8-s − 0.726·9-s − 0.351·10-s + 0.949·11-s − 0.261·12-s − 1.69·13-s − 0.489·14-s + 0.259·15-s + 0.250·16-s + 0.322·17-s − 0.513·18-s + 0.357·19-s − 0.248·20-s + 0.362·21-s + 0.671·22-s + 1.09·23-s − 0.184·24-s − 0.753·25-s − 1.20·26-s + 0.902·27-s − 0.346·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.583689857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583689857\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 0.905T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 1.83T + 7T^{2} \) |
| 11 | \( 1 - 3.15T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 - 1.55T + 19T^{2} \) |
| 23 | \( 1 - 5.24T + 23T^{2} \) |
| 29 | \( 1 + 1.90T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 + 8.79T + 37T^{2} \) |
| 41 | \( 1 + 1.96T + 41T^{2} \) |
| 43 | \( 1 + 2.51T + 43T^{2} \) |
| 47 | \( 1 - 7.53T + 47T^{2} \) |
| 53 | \( 1 - 2.37T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 2.35T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 - 7.62T + 73T^{2} \) |
| 79 | \( 1 - 5.41T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 + 0.906T + 97T^{2} \) |
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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
−8.371588449418231805434829187048, −7.41809129512854385948878429291, −6.87983923418942001024518604366, −6.22242579366678102251066452291, −5.31379602503213453232078205706, −4.86002929072688797820778899024, −3.78760143616218145993454367401, −3.15440521587436397452579683643, −2.20689551074270416164976582325, −0.63881465649874637869000546485,
0.63881465649874637869000546485, 2.20689551074270416164976582325, 3.15440521587436397452579683643, 3.78760143616218145993454367401, 4.86002929072688797820778899024, 5.31379602503213453232078205706, 6.22242579366678102251066452291, 6.87983923418942001024518604366, 7.41809129512854385948878429291, 8.371588449418231805434829187048
