| L(s) = 1 | + i·3-s + i·7-s − 9-s − 4.44·11-s − 2.44i·13-s − 3.44i·17-s − 7.34·19-s − 21-s − i·23-s − i·27-s + 9.44·29-s − 1.89·31-s − 4.44i·33-s − 9.89i·37-s + 2.44·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.377i·7-s − 0.333·9-s − 1.34·11-s − 0.679i·13-s − 0.836i·17-s − 1.68·19-s − 0.218·21-s − 0.208i·23-s − 0.192i·27-s + 1.75·29-s − 0.341·31-s − 0.774i·33-s − 1.62i·37-s + 0.392·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.279956711\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.279956711\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - iT - 7T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + 3.44iT - 17T^{2} \) |
| 19 | \( 1 + 7.34T + 19T^{2} \) |
| 29 | \( 1 - 9.44T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 + 9.89iT - 37T^{2} \) |
| 41 | \( 1 + 0.550T + 41T^{2} \) |
| 43 | \( 1 - 7.79iT - 43T^{2} \) |
| 47 | \( 1 - 7.34iT - 47T^{2} \) |
| 53 | \( 1 + 4.34iT - 53T^{2} \) |
| 59 | \( 1 - 6.55T + 59T^{2} \) |
| 61 | \( 1 - 0.449T + 61T^{2} \) |
| 67 | \( 1 - 8.79iT - 67T^{2} \) |
| 71 | \( 1 - 2.34T + 71T^{2} \) |
| 73 | \( 1 - 7.34iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 - 15.4iT - 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
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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
−8.250218229261969247986287211029, −7.48798000741704326785257263251, −6.61695438552252573796770840184, −5.85163458118044833736888380961, −5.19185113491397915728215025493, −4.61442598403851442222482269322, −3.76817378375079348899863129121, −2.64597741395011625175523822498, −2.42503599151958582720818296757, −0.68502340513980062447050138675,
0.45263153025545721513988502309, 1.75624935628303556591469966681, 2.40632371359559877412702427040, 3.36994151581093976930997991288, 4.31191390854305006432098092524, 4.94991970835248592635296359697, 5.85823240860611581496770586435, 6.55181880838343658152915948167, 7.03794079556070990732760205121, 7.947069103211280545808572967565
