| L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.434 − 0.158i)5-s + (−2.12 − 3.67i)7-s + (0.173 + 0.984i)9-s + (2.54 − 4.40i)11-s + (−3.32 + 2.79i)13-s + (−0.434 − 0.158i)15-s + (1.12 − 6.36i)17-s + (4.31 − 0.582i)19-s + (−0.736 + 4.17i)21-s + (−1.45 − 0.530i)23-s + (−3.66 + 3.07i)25-s + (0.500 − 0.866i)27-s + (−0.495 − 2.81i)29-s + (2.16 + 3.75i)31-s + ⋯ |
| L(s) = 1 | + (−0.442 − 0.371i)3-s + (0.194 − 0.0707i)5-s + (−0.801 − 1.38i)7-s + (0.0578 + 0.328i)9-s + (0.767 − 1.32i)11-s + (−0.922 + 0.774i)13-s + (−0.112 − 0.0408i)15-s + (0.272 − 1.54i)17-s + (0.991 − 0.133i)19-s + (−0.160 + 0.911i)21-s + (−0.304 − 0.110i)23-s + (−0.733 + 0.615i)25-s + (0.0962 − 0.166i)27-s + (−0.0920 − 0.522i)29-s + (0.389 + 0.674i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0562 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0562 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.644429 - 0.681733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.644429 - 0.681733i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-4.31 + 0.582i)T \) |
| good | 5 | \( 1 + (-0.434 + 0.158i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (2.12 + 3.67i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.54 + 4.40i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.32 - 2.79i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.12 + 6.36i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (1.45 + 0.530i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.495 + 2.81i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.16 - 3.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 3.48T + 37T^{2} \) |
| 41 | \( 1 + (-8.79 - 7.38i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.07 - 1.84i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.86 - 10.5i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-8.22 - 2.99i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.14 + 12.1i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.32 - 1.21i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.752 + 4.26i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.89 - 2.14i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.545 + 0.458i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (5.21 + 4.37i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.390 - 0.675i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.56 + 1.31i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (0.104 - 0.594i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.76810442278251368301598556977, −11.26017856682375527298108377186, −9.906281724894661124879697146105, −9.340665579034680155156786362507, −7.67308536205677618132761624765, −6.93175183288794701167562972150, −5.94824262825145657656395682309, −4.51873200128923544583342028315, −3.14605483390952207479617959225, −0.847273641629958406918971337733,
2.30530166902156775601991626276, 3.90153711537805315508191602774, 5.39427100140721612453571451956, 6.12047382375364826343536300354, 7.37169111505660352364627790090, 8.770095517804136253840763848884, 9.800493480995298606842284977134, 10.19195773231325365278641782628, 11.90596928681410617272834246022, 12.20560791338646521422010411109
