L(s) = 1 | + (0.587 + 0.809i)2-s + (−0.309 + 0.951i)4-s + (1.87 + 1.22i)5-s + 3.26i·7-s + (−0.951 + 0.309i)8-s + (0.107 + 2.23i)10-s + (−1.44 + 1.05i)11-s + (3.38 − 4.65i)13-s + (−2.63 + 1.91i)14-s + (−0.809 − 0.587i)16-s + (−6.26 + 2.03i)17-s + (−1.21 − 3.74i)19-s + (−1.74 + 1.39i)20-s + (−1.70 − 0.552i)22-s + (4.91 + 6.76i)23-s + ⋯ |
L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.154 + 0.475i)4-s + (0.836 + 0.548i)5-s + 1.23i·7-s + (−0.336 + 0.109i)8-s + (0.0340 + 0.706i)10-s + (−0.436 + 0.317i)11-s + (0.938 − 1.29i)13-s + (−0.705 + 0.512i)14-s + (−0.202 − 0.146i)16-s + (−1.51 + 0.493i)17-s + (−0.278 − 0.858i)19-s + (−0.389 + 0.313i)20-s + (−0.362 − 0.117i)22-s + (1.02 + 1.41i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15602 + 1.44605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15602 + 1.44605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.587 - 0.809i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.87 - 1.22i)T \) |
good | 7 | \( 1 - 3.26iT - 7T^{2} \) |
| 11 | \( 1 + (1.44 - 1.05i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-3.38 + 4.65i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (6.26 - 2.03i)T + (13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.21 + 3.74i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.91 - 6.76i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.442 + 1.36i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.99 - 6.15i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.56 + 9.03i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (3.18 + 2.31i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.18iT - 43T^{2} \) |
| 47 | \( 1 + (-6.88 - 2.23i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.47 + 0.478i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (8.85 + 6.43i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.637 + 0.463i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-7.46 + 2.42i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.12 + 6.55i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (7.26 + 10.0i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.18 + 3.64i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.37 + 2.07i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.29 - 0.939i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-8.68 - 2.82i)T + (78.4 + 57.0i)T^{2} \) |
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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
−11.19627154221132515443262515806, −10.62841664071569293942601210353, −9.251265176788330982376506360433, −8.749599169969058300382837460411, −7.55194354761330188950446011507, −6.45858347016868544937436799368, −5.75373396117021537422881917471, −4.95121476614419616125579945889, −3.26570910727595805260851869726, −2.24906097889226931330420998950,
1.10839608211035326118421000535, 2.49598524934884033806797302694, 4.13694982828818824593910750658, 4.70427329653420423943749160153, 6.15610948835004018717799036872, 6.80725823925347408244019094975, 8.363763285402486792848582621055, 9.150990328408903473656827752875, 10.12439298428478929860591153628, 10.85987478913232281182279832010