| L(s) = 1 | + i·2-s − 4-s + (−2.21 − 0.311i)5-s + 4.42i·7-s − i·8-s + (0.311 − 2.21i)10-s − 2.62·11-s + 5.80i·13-s − 4.42·14-s + 16-s − 3.80i·17-s − 19-s + (2.21 + 0.311i)20-s − 2.62i·22-s + 2.62i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.990 − 0.139i)5-s + 1.67i·7-s − 0.353i·8-s + (0.0983 − 0.700i)10-s − 0.790·11-s + 1.61i·13-s − 1.18·14-s + 0.250·16-s − 0.923i·17-s − 0.229·19-s + (0.495 + 0.0695i)20-s − 0.559i·22-s + 0.546i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.139 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1885972463\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1885972463\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.21 + 0.311i)T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 4.42iT - 7T^{2} \) |
| 11 | \( 1 + 2.62T + 11T^{2} \) |
| 13 | \( 1 - 5.80iT - 13T^{2} \) |
| 17 | \( 1 + 3.80iT - 17T^{2} \) |
| 23 | \( 1 - 2.62iT - 23T^{2} \) |
| 29 | \( 1 - 3.37T + 29T^{2} \) |
| 31 | \( 1 + 4.42T + 31T^{2} \) |
| 37 | \( 1 + 5.80iT - 37T^{2} \) |
| 41 | \( 1 + 5.67T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 - 2.62iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 1.05T + 59T^{2} \) |
| 61 | \( 1 - 4.75T + 61T^{2} \) |
| 67 | \( 1 + 15.6iT - 67T^{2} \) |
| 71 | \( 1 + 15.6T + 71T^{2} \) |
| 73 | \( 1 - 11.6iT - 73T^{2} \) |
| 79 | \( 1 + 4.42T + 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.37iT - 97T^{2} \) |
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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
−9.458906066969828320744934173273, −8.958476863157749918690001103255, −8.415975982130670727157621872153, −7.46748080219260903312997463320, −6.86183727322038432955913592777, −5.80671401757700725310564011779, −5.10666948806117520161437669204, −4.32374928898442614956378691707, −3.16133660618818033310383816337, −2.03967464748216691122344665435,
0.080419635018876363814260250089, 1.14406286209265805409309546206, 2.86331760585207343634621633809, 3.58151890875295558672073871279, 4.35807294499357119703970615030, 5.17046140775741018035150818397, 6.45587150527733816690945806484, 7.42085237999644769594620475241, 8.029610239424787246804078335945, 8.510126743684956437479174357062
