L(s) = 1 | + (−0.570 + 1.29i)2-s + (−0.637 + 1.10i)3-s + (−1.34 − 1.47i)4-s + (−1.60 + 2.77i)5-s + (−1.06 − 1.45i)6-s − 1.25i·7-s + (2.68 − 0.903i)8-s + (0.688 + 1.19i)9-s + (−2.67 − 3.65i)10-s + 2.11i·11-s + (2.48 − 0.548i)12-s + (2.12 − 1.22i)13-s + (1.61 + 0.713i)14-s + (−2.04 − 3.53i)15-s + (−0.359 + 3.98i)16-s + (0.765 − 1.32i)17-s + ⋯ |
L(s) = 1 | + (−0.403 + 0.915i)2-s + (−0.367 + 0.637i)3-s + (−0.674 − 0.738i)4-s + (−0.717 + 1.24i)5-s + (−0.434 − 0.593i)6-s − 0.472i·7-s + (0.947 − 0.319i)8-s + (0.229 + 0.397i)9-s + (−0.847 − 1.15i)10-s + 0.636i·11-s + (0.718 − 0.158i)12-s + (0.590 − 0.341i)13-s + (0.432 + 0.190i)14-s + (−0.527 − 0.913i)15-s + (−0.0898 + 0.995i)16-s + (0.185 − 0.321i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 - 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.219630 + 0.568028i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.219630 + 0.568028i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.570 - 1.29i)T \) |
| 19 | \( 1 + (-3.76 - 2.19i)T \) |
good | 3 | \( 1 + (0.637 - 1.10i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.60 - 2.77i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 1.25iT - 7T^{2} \) |
| 11 | \( 1 - 2.11iT - 11T^{2} \) |
| 13 | \( 1 + (-2.12 + 1.22i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.765 + 1.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-7.61 + 4.39i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.20 - 3.00i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 7.78T + 31T^{2} \) |
| 37 | \( 1 + 9.97iT - 37T^{2} \) |
| 41 | \( 1 + (-1.09 - 0.631i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.04 - 2.91i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (6.12 - 3.53i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.18 + 3.57i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.83 + 4.91i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.80 - 4.86i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.0235 - 0.0408i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.12 + 5.41i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.658 + 1.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.77 + 6.53i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 7.84iT - 83T^{2} \) |
| 89 | \( 1 + (6.02 - 3.47i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.51 - 4.91i)T + (48.5 + 84.0i)T^{2} \) |
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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
−15.01070049284793904857400495220, −14.39850890623825995859675369707, −12.99657771271881127402266553819, −11.02786890518423636830239468118, −10.61423470723515621526985400573, −9.375563948707857963428711322275, −7.64116387022858394721158913585, −7.02519332486971638142933579922, −5.35920870027863150556404240307, −3.86793961139571033658096874209,
1.11617401621927805996136514329, 3.70681872606563556368126385263, 5.35760814191215756760134593172, 7.37712462993568658443695382619, 8.645630780802033441784910443729, 9.373727797071829921253560828468, 11.26607158441047197968305318698, 11.82865401266075169743663248897, 12.82381137144410725252720253891, 13.43580994730627728721725549894