| L(s) = 1 | + (1.27 + 1.17i)3-s + (0.0952 + 0.0346i)5-s + (0.165 − 0.935i)7-s + (0.240 + 2.99i)9-s + (−3.80 + 1.38i)11-s + (3.80 − 3.18i)13-s + (0.0805 + 0.156i)15-s + (−3.11 − 5.39i)17-s + (−0.514 + 0.891i)19-s + (1.30 − 0.997i)21-s + (0.602 + 3.41i)23-s + (−3.82 − 3.20i)25-s + (−3.20 + 4.08i)27-s + (−3.04 − 2.55i)29-s + (−0.740 − 4.20i)31-s + ⋯ |
| L(s) = 1 | + (0.734 + 0.678i)3-s + (0.0426 + 0.0155i)5-s + (0.0623 − 0.353i)7-s + (0.0800 + 0.996i)9-s + (−1.14 + 0.417i)11-s + (1.05 − 0.884i)13-s + (0.0207 + 0.0402i)15-s + (−0.755 − 1.30i)17-s + (−0.118 + 0.204i)19-s + (0.285 − 0.217i)21-s + (0.125 + 0.712i)23-s + (−0.764 − 0.641i)25-s + (−0.617 + 0.786i)27-s + (−0.566 − 0.475i)29-s + (−0.133 − 0.754i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18849 + 0.302434i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18849 + 0.302434i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.27 - 1.17i)T \) |
| good | 5 | \( 1 + (-0.0952 - 0.0346i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.165 + 0.935i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (3.80 - 1.38i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.80 + 3.18i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.11 + 5.39i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.514 - 0.891i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.602 - 3.41i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.04 + 2.55i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.740 + 4.20i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.19 - 7.26i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.15 + 1.80i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (6.32 - 2.30i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.35 - 7.70i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 + (-13.3 - 4.87i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.469 - 2.66i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.53 - 2.12i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.26 + 12.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.981 + 0.823i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-4.02 - 3.37i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.80 - 8.32i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.70 - 1.34i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.50187624227835799667865731486, −13.32217323333768592547947251544, −11.47849941047860201464216943669, −10.46061941600643053738127661342, −9.618441107205387439022740791069, −8.336847433399346199058213465767, −7.45137679896732830357767738992, −5.55950735387134248025417594622, −4.21258775893734228130369701571, −2.68823161421162468823525613198,
2.09869540012092104113038302604, 3.79468097467616336744615450026, 5.79039086587861507248858220749, 6.99274644795809887802054813492, 8.359767011992442202216852467424, 8.925808124413771608342733566680, 10.50701451585169322057752629039, 11.61493721812460054310800906294, 12.97275003847108495852102049518, 13.38080880725989638721176709257
