| L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + (2.51 + 4.32i)5-s − 2.44i·6-s + (−0.686 + 6.96i)7-s − 2.82i·8-s + (−1.5 + 2.59i)9-s + (−0.0171 − 7.07i)10-s + (−2.20 − 3.82i)11-s + (−1.73 + 2.99i)12-s − 19.3·13-s + (5.76 − 8.04i)14-s + (−4.31 + 7.51i)15-s + (−2.00 + 3.46i)16-s + (1.48 + 2.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + (0.502 + 0.864i)5-s − 0.408i·6-s + (−0.0980 + 0.995i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.00171 − 0.707i)10-s + (−0.200 − 0.347i)11-s + (−0.144 + 0.249i)12-s − 1.49·13-s + (0.411 − 0.574i)14-s + (−0.287 + 0.500i)15-s + (−0.125 + 0.216i)16-s + (0.0872 + 0.151i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.631573 + 0.857464i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.631573 + 0.857464i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 + (-2.51 - 4.32i)T \) |
| 7 | \( 1 + (0.686 - 6.96i)T \) |
| good | 11 | \( 1 + (2.20 + 3.82i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 19.3T + 169T^{2} \) |
| 17 | \( 1 + (-1.48 - 2.57i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-2.06 - 1.19i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-1.29 - 0.748i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 39.2T + 841T^{2} \) |
| 31 | \( 1 + (35.0 - 20.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-36.8 - 21.2i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 57.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-38.5 + 66.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-81.3 + 46.9i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (8.24 - 4.75i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (35.0 + 20.2i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-53.1 + 30.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 46.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-1.76 - 3.06i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-47.5 + 82.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 94.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-133. - 76.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 122.T + 9.40e3T^{2} \) |
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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
−12.19435952119107259036478454669, −11.31907034088199565565478824885, −10.23037802196237302269449908715, −9.675140198982284190135709367536, −8.690256595057293576560025933262, −7.56495398098587500677041426997, −6.32710884897322570596755023256, −5.03760981969561349162317308027, −3.15624556623503106356625414011, −2.29819166791837180449318607506,
0.68309609931707223443209330570, 2.29291974492680187690085908769, 4.43716201701663064187582025382, 5.67697283876597242707169721482, 7.09715426212710461639042051487, 7.67052233719700016702716589777, 8.907023059763771139369939603922, 9.732905624284098467422335739914, 10.55901276993134230044987045853, 12.04226856276589362134587420734
