L(s) = 1 | + (−0.841 + 0.540i)2-s + (0.415 − 0.909i)4-s + (−0.186 + 0.0547i)5-s + (−1.27 − 1.47i)7-s + (0.142 + 0.989i)8-s + (0.127 − 0.146i)10-s + (−5.30 − 3.40i)11-s + (−3.15 + 3.63i)13-s + (1.87 + 0.549i)14-s + (−0.654 − 0.755i)16-s + (1.33 + 2.91i)17-s + (0.357 − 0.781i)19-s + (−0.0276 + 0.192i)20-s + 6.30·22-s + (−3.37 − 3.40i)23-s + ⋯ |
L(s) = 1 | + (−0.594 + 0.382i)2-s + (0.207 − 0.454i)4-s + (−0.0833 + 0.0244i)5-s + (−0.483 − 0.557i)7-s + (0.0503 + 0.349i)8-s + (0.0402 − 0.0464i)10-s + (−1.59 − 1.02i)11-s + (−0.874 + 1.00i)13-s + (0.500 + 0.146i)14-s + (−0.163 − 0.188i)16-s + (0.323 + 0.707i)17-s + (0.0819 − 0.179i)19-s + (−0.00618 + 0.0429i)20-s + 1.34·22-s + (−0.704 − 0.709i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.563i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0500001 - 0.162157i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0500001 - 0.162157i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (3.37 + 3.40i)T \) |
good | 5 | \( 1 + (0.186 - 0.0547i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (1.27 + 1.47i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (5.30 + 3.40i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (3.15 - 3.63i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.33 - 2.91i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.357 + 0.781i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.25 + 4.92i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.144 - 1.00i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (6.61 + 1.94i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (0.932 - 0.273i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.731 + 5.09i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 7.41T + 47T^{2} \) |
| 53 | \( 1 + (-1.31 - 1.51i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.63 - 6.50i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.72 + 11.9i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-5.96 + 3.83i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (0.933 - 0.599i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (5.47 - 11.9i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (0.739 - 0.853i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.84 - 1.12i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.35 - 9.45i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-1.07 + 0.315i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.55910933640126210446271732633, −10.05244968894610364723557869471, −8.979828724429986883897347628617, −7.996966495088000861993468237533, −7.30114001831282192648500147036, −6.20297558095337865468293836849, −5.23787438351380780154578569683, −3.80663306102771757322678215173, −2.29650531618561966296893804459, −0.12071622720488404338253531590,
2.22609868106672032291541574105, 3.17602857170913053762281366897, 4.86807065736210396449489619543, 5.79845070873265628858532083305, 7.41570270144840824480845108338, 7.74926847939548934092045396243, 9.001789712037036467026113700252, 10.07092158645733800232135946438, 10.24509069546799548555825378990, 11.62567100371440174861387807682