| L(s) = 1 | + (0.786 − 0.618i)2-s + (−1.59 − 2.23i)3-s + (0.235 − 0.971i)4-s + (−3.72 − 0.717i)5-s + (−2.63 − 0.773i)6-s + (1.75 + 1.97i)7-s + (−0.415 − 0.909i)8-s + (−1.48 + 4.29i)9-s + (−3.36 + 1.73i)10-s + (−0.867 − 0.682i)11-s + (−2.54 + 1.02i)12-s + (−1.38 − 0.892i)13-s + (2.60 + 0.465i)14-s + (4.32 + 9.46i)15-s + (−0.888 − 0.458i)16-s + (0.703 − 0.670i)17-s + ⋯ |
| L(s) = 1 | + (0.555 − 0.437i)2-s + (−0.919 − 1.29i)3-s + (0.117 − 0.485i)4-s + (−1.66 − 0.320i)5-s + (−1.07 − 0.315i)6-s + (0.664 + 0.746i)7-s + (−0.146 − 0.321i)8-s + (−0.494 + 1.43i)9-s + (−1.06 + 0.549i)10-s + (−0.261 − 0.205i)11-s + (−0.735 + 0.294i)12-s + (−0.385 − 0.247i)13-s + (0.696 + 0.124i)14-s + (1.11 + 2.44i)15-s + (−0.222 − 0.114i)16-s + (0.170 − 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 - 0.670i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.174010 + 0.452417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.174010 + 0.452417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.786 + 0.618i)T \) |
| 7 | \( 1 + (-1.75 - 1.97i)T \) |
| 23 | \( 1 + (4.10 - 2.47i)T \) |
| good | 3 | \( 1 + (1.59 + 2.23i)T + (-0.981 + 2.83i)T^{2} \) |
| 5 | \( 1 + (3.72 + 0.717i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (0.867 + 0.682i)T + (2.59 + 10.6i)T^{2} \) |
| 13 | \( 1 + (1.38 + 0.892i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.703 + 0.670i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (1.00 + 0.956i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (7.10 + 2.08i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (4.64 + 0.443i)T + (30.4 + 5.86i)T^{2} \) |
| 37 | \( 1 + (-1.99 + 5.77i)T + (-29.0 - 22.8i)T^{2} \) |
| 41 | \( 1 + (-7.36 + 8.49i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.32 + 5.09i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + (5.20 + 9.01i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0304 + 0.639i)T + (-52.7 + 5.03i)T^{2} \) |
| 59 | \( 1 + (11.5 - 5.93i)T + (34.2 - 48.0i)T^{2} \) |
| 61 | \( 1 + (0.0826 - 0.116i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-6.85 - 2.74i)T + (48.4 + 46.2i)T^{2} \) |
| 71 | \( 1 + (2.17 + 15.1i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (3.12 - 12.8i)T + (-64.8 - 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.602 + 12.6i)T + (-78.6 - 7.50i)T^{2} \) |
| 83 | \( 1 + (-8.85 - 10.2i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-7.74 + 0.739i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-7.53 + 8.69i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.41179600079311603322062801104, −10.82938308433534112797016622828, −8.988358619753866142600557837841, −7.75606028333882833486599154775, −7.39050014688156931220154340653, −5.89514635607925690878710716681, −5.13034374264858142981607101603, −3.83506352594609888336065140627, −2.07287563766777533737354800502, −0.31711424857321746314325100208,
3.48954268403528404976406654293, 4.33821322391943778598254711143, 4.82418255791197142287659732605, 6.21279395916905961194767307665, 7.47504480764172973395717643965, 8.072506274426469940595057400096, 9.590977955915286136482499809803, 10.78759287946224886464118711634, 11.17400587372977253007450391073, 11.92954346919790597009961665099
