L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + 2.13·5-s + (0.499 + 0.866i)6-s + (0.0244 + 0.0423i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−1.06 + 1.85i)10-s + (−3.14 + 5.45i)11-s − 0.999·12-s − 0.0489·14-s + (1.06 − 1.85i)15-s + (−0.5 + 0.866i)16-s + (1.44 + 2.50i)17-s + 0.999·18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 − 0.433i)4-s + 0.955·5-s + (0.204 + 0.353i)6-s + (0.00924 + 0.0160i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.337 + 0.585i)10-s + (−0.949 + 1.64i)11-s − 0.288·12-s − 0.0130·14-s + (0.275 − 0.477i)15-s + (−0.125 + 0.216i)16-s + (0.350 + 0.607i)17-s + 0.235·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.540947880\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.540947880\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 + (-0.0244 - 0.0423i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.14 - 5.45i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.44 - 2.50i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.60 - 6.24i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.35 - 2.35i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.45 - 4.25i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.00T + 31T^{2} \) |
| 37 | \( 1 + (-0.0881 + 0.152i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.29 + 7.44i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.35 + 5.81i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.20T + 47T^{2} \) |
| 53 | \( 1 - 9.34T + 53T^{2} \) |
| 59 | \( 1 + (-2.13 - 3.69i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.55 + 6.15i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.35 - 7.54i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + (1.96 - 3.39i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.23 - 2.14i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.14828183073464934549518102098, −9.319421395349733314642573668164, −8.316886756337322341108635767484, −7.55445305661540972612685234032, −6.96297634650170102556273268195, −5.79034614191833313582021489480, −5.34855208563983390349418170343, −3.96984261906793910970053790304, −2.38557592525018930746611003799, −1.51647143752893301238846409288,
0.800666923925689886175715814433, 2.59056314351572235615996551960, 3.00186018764922244834283016826, 4.44248264905973625525149944220, 5.42263565373829283681010674924, 6.19691396962317363118670215549, 7.58149415901100981929457795471, 8.355138647610985296393409205903, 9.173921057417889857844978120331, 9.786274202528519929772605173197