| L(s) = 1 | + 2.73·3-s − 5-s + 3.93·7-s + 4.46·9-s + 3.17·11-s + 6.66·13-s − 2.73·15-s − 17-s + 5.46·19-s + 10.7·21-s − 8.81·23-s + 25-s + 3.99·27-s − 3.20·29-s − 5.83·31-s + 8.66·33-s − 3.93·35-s − 4.07·37-s + 18.2·39-s + 0.799·41-s − 8.30·43-s − 4.46·45-s − 6.48·47-s + 8.46·49-s − 2.73·51-s − 2.87·53-s − 3.17·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.447·5-s + 1.48·7-s + 1.48·9-s + 0.956·11-s + 1.84·13-s − 0.705·15-s − 0.242·17-s + 1.25·19-s + 2.34·21-s − 1.83·23-s + 0.200·25-s + 0.769·27-s − 0.594·29-s − 1.04·31-s + 1.50·33-s − 0.664·35-s − 0.670·37-s + 2.91·39-s + 0.124·41-s − 1.26·43-s − 0.665·45-s − 0.946·47-s + 1.20·49-s − 0.382·51-s − 0.395·53-s − 0.427·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.035256993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.035256993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 11 | \( 1 - 3.17T + 11T^{2} \) |
| 13 | \( 1 - 6.66T + 13T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 + 8.81T + 23T^{2} \) |
| 29 | \( 1 + 3.20T + 29T^{2} \) |
| 31 | \( 1 + 5.83T + 31T^{2} \) |
| 37 | \( 1 + 4.07T + 37T^{2} \) |
| 41 | \( 1 - 0.799T + 41T^{2} \) |
| 43 | \( 1 + 8.30T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 + 1.67T + 73T^{2} \) |
| 79 | \( 1 + 0.371T + 79T^{2} \) |
| 83 | \( 1 + 5.02T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.754175634982520816781774380393, −8.005855012066898981501211017373, −7.81292040740142612219440838679, −6.75587021963571152066520794132, −5.73519859261647588892284513416, −4.63074753667523668702700236696, −3.72180624897142015616246856261, −3.45541048027171863961632900752, −1.90398930912352028317400143486, −1.44138172109481042156902524397,
1.44138172109481042156902524397, 1.90398930912352028317400143486, 3.45541048027171863961632900752, 3.72180624897142015616246856261, 4.63074753667523668702700236696, 5.73519859261647588892284513416, 6.75587021963571152066520794132, 7.81292040740142612219440838679, 8.005855012066898981501211017373, 8.754175634982520816781774380393
