| L(s) = 1 | − 2.44i·3-s − 0.200i·5-s − 2.34·7-s − 2.99·9-s + 1.63i·13-s − 0.490·15-s + 3.33·17-s − 4.92i·19-s + 5.74i·21-s + 9.25·23-s + 4.95·25-s − 0.0187i·27-s + 6.19i·29-s + 8.25·31-s + 0.470i·35-s + ⋯ |
| L(s) = 1 | − 1.41i·3-s − 0.0896i·5-s − 0.886·7-s − 0.997·9-s + 0.452i·13-s − 0.126·15-s + 0.809·17-s − 1.12i·19-s + 1.25i·21-s + 1.92·23-s + 0.991·25-s − 0.00360i·27-s + 1.14i·29-s + 1.48·31-s + 0.0795i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.735273393\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.735273393\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.44iT - 3T^{2} \) |
| 5 | \( 1 + 0.200iT - 5T^{2} \) |
| 7 | \( 1 + 2.34T + 7T^{2} \) |
| 13 | \( 1 - 1.63iT - 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 4.92iT - 19T^{2} \) |
| 23 | \( 1 - 9.25T + 23T^{2} \) |
| 29 | \( 1 - 6.19iT - 29T^{2} \) |
| 31 | \( 1 - 8.25T + 31T^{2} \) |
| 37 | \( 1 - 1.28iT - 37T^{2} \) |
| 41 | \( 1 - 6.03T + 41T^{2} \) |
| 43 | \( 1 + 2.47iT - 43T^{2} \) |
| 47 | \( 1 + 4.75T + 47T^{2} \) |
| 53 | \( 1 + 8.76iT - 53T^{2} \) |
| 59 | \( 1 + 1.90iT - 59T^{2} \) |
| 61 | \( 1 - 6.39iT - 61T^{2} \) |
| 67 | \( 1 + 8.27iT - 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + 14.6T + 79T^{2} \) |
| 83 | \( 1 - 9.37iT - 83T^{2} \) |
| 89 | \( 1 - 5.11T + 89T^{2} \) |
| 97 | \( 1 - 0.299T + 97T^{2} \) |
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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
−8.199992100355681390658070289998, −7.20072061904479536738483201083, −6.88150340957450235821382834622, −6.36226811192891505175994701007, −5.34393032233311443490062718877, −4.56648536146156241677488966266, −3.15365183197612233377950750041, −2.74650799214331671168817715862, −1.43592350844067539980809300216, −0.64651758553240247448092062488,
1.01792149932073417670137532645, 2.86398880180400874648921069394, 3.21287246999243114949563357912, 4.16326052534573386235827978771, 4.84657244703713285074725600707, 5.67213612949106150729717921875, 6.32731908730038543500743171228, 7.27397437679017290417707601487, 8.121605156716974550802924286271, 8.927927460168701049133121448405
