L(s) = 1 | − 2-s + 4-s + (0.724 + 1.25i)5-s − 8-s + (−0.724 − 1.25i)10-s + (1 − 1.73i)11-s + (2.44 − 4.24i)13-s + 16-s + (−1 − 1.73i)17-s + (1.27 − 2.20i)19-s + (0.724 + 1.25i)20-s + (−1 + 1.73i)22-s + (−0.5 − 0.866i)23-s + (1.44 − 2.51i)25-s + (−2.44 + 4.24i)26-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (0.324 + 0.561i)5-s − 0.353·8-s + (−0.229 − 0.396i)10-s + (0.301 − 0.522i)11-s + (0.679 − 1.17i)13-s + 0.250·16-s + (−0.242 − 0.420i)17-s + (0.292 − 0.506i)19-s + (0.162 + 0.280i)20-s + (−0.213 + 0.369i)22-s + (−0.104 − 0.180i)23-s + (0.289 − 0.502i)25-s + (−0.480 + 0.832i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9474298727\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9474298727\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.724 - 1.25i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1 + 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.44 + 4.24i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.27 + 2.20i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.44 + 5.97i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + (5.89 - 10.2i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.89 - 8.48i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 + 5.97i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + (5.44 + 9.43i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2T + 59T^{2} \) |
| 61 | \( 1 + 6.55T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 + 0.101T + 71T^{2} \) |
| 73 | \( 1 + (3.44 + 5.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-8.44 + 14.6i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.44 - 2.51i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.663357434513120060847198780311, −7.985831225001440066777383192477, −7.18691458308647969229879594083, −6.38995667115646391692715433815, −5.81578362272830370391322064065, −4.81116534970986262043132583878, −3.46458638948626067155197858001, −2.87415436333564596489701880808, −1.68580527057421435156772691276, −0.38830461814407498132323967983,
1.43288977544691183617403259661, 1.92140828772393633274759738553, 3.44374718276081407306528964037, 4.23899127375800743890859745677, 5.34955447816144983921965212469, 6.02472323959520825532774661001, 7.02860151417438849610375922216, 7.45476625238191775216299801247, 8.616654250294836651946828935073, 9.126523638661937039606718447525