| L(s) = 1 | + 2-s + 3-s + 4-s + 3.56·5-s + 6-s − 7-s + 8-s + 9-s + 3.56·10-s − 1.56·11-s + 12-s + 0.438·13-s − 14-s + 3.56·15-s + 16-s + 17-s + 18-s − 7.12·19-s + 3.56·20-s − 21-s − 1.56·22-s + 3.12·23-s + 24-s + 7.68·25-s + 0.438·26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.59·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 1.12·10-s − 0.470·11-s + 0.288·12-s + 0.121·13-s − 0.267·14-s + 0.919·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s − 1.63·19-s + 0.796·20-s − 0.218·21-s − 0.332·22-s + 0.651·23-s + 0.204·24-s + 1.53·25-s + 0.0859·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.331587077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.331587077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 43 | \( 1 + 6.43T + 43T^{2} \) |
| 47 | \( 1 - 6.24T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 8.68T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 2.68T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−10.46033872160948618844677999938, −9.527240261047846071594435780035, −8.929191848346231019438514474748, −7.71162845078776967273320954564, −6.65845283599482752822200977814, −5.91930839288568997793257076134, −5.09551303775463350740694175216, −3.83334930195752607830870227671, −2.63284490762236157385097736977, −1.82646557097927548125837226914,
1.82646557097927548125837226914, 2.63284490762236157385097736977, 3.83334930195752607830870227671, 5.09551303775463350740694175216, 5.91930839288568997793257076134, 6.65845283599482752822200977814, 7.71162845078776967273320954564, 8.929191848346231019438514474748, 9.527240261047846071594435780035, 10.46033872160948618844677999938
