| L(s) = 1 | − 2.68·3-s − 5-s + 7-s + 4.22·9-s − 13-s + 2.68·15-s + 1.22·17-s − 0.777·19-s − 2.68·21-s + 7.81·23-s + 25-s − 3.28·27-s − 7.13·29-s + 1.31·31-s − 35-s − 8.50·37-s + 2.68·39-s + 1.22·41-s + 1.37·43-s − 4.22·45-s − 3.81·47-s + 49-s − 3.28·51-s + 2·53-s + 2.09·57-s − 4.06·59-s − 10·61-s + ⋯ |
| L(s) = 1 | − 1.55·3-s − 0.447·5-s + 0.377·7-s + 1.40·9-s − 0.277·13-s + 0.693·15-s + 0.296·17-s − 0.178·19-s − 0.586·21-s + 1.63·23-s + 0.200·25-s − 0.632·27-s − 1.32·29-s + 0.235·31-s − 0.169·35-s − 1.39·37-s + 0.430·39-s + 0.190·41-s + 0.209·43-s − 0.629·45-s − 0.557·47-s + 0.142·49-s − 0.459·51-s + 0.274·53-s + 0.276·57-s − 0.528·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8006221169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8006221169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 - 1.22T + 17T^{2} \) |
| 19 | \( 1 + 0.777T + 19T^{2} \) |
| 23 | \( 1 - 7.81T + 23T^{2} \) |
| 29 | \( 1 + 7.13T + 29T^{2} \) |
| 31 | \( 1 - 1.31T + 31T^{2} \) |
| 37 | \( 1 + 8.50T + 37T^{2} \) |
| 41 | \( 1 - 1.22T + 41T^{2} \) |
| 43 | \( 1 - 1.37T + 43T^{2} \) |
| 47 | \( 1 + 3.81T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 4.06T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 5.37T + 71T^{2} \) |
| 73 | \( 1 + 0.930T + 73T^{2} \) |
| 79 | \( 1 - 0.777T + 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 - 19.0T + 97T^{2} \) |
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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
−8.524292984173163514023410396298, −7.54312165642481378372799501941, −7.02563063596333029396700723542, −6.27671680147623795093464811242, −5.38030660287290178254403216167, −4.99364018922852794547145236527, −4.14134494086035703470492591008, −3.11323311213912354113152463034, −1.68587159636540516011195790087, −0.58017825424418269792892065228,
0.58017825424418269792892065228, 1.68587159636540516011195790087, 3.11323311213912354113152463034, 4.14134494086035703470492591008, 4.99364018922852794547145236527, 5.38030660287290178254403216167, 6.27671680147623795093464811242, 7.02563063596333029396700723542, 7.54312165642481378372799501941, 8.524292984173163514023410396298
