| L(s) = 1 | − 2i·3-s + (1 + 2i)5-s − 9-s − 4·11-s − 4i·13-s + (4 − 2i)15-s − 4·19-s − 2i·23-s + (−3 + 4i)25-s − 4i·27-s − 2·29-s + 8i·33-s − 4i·37-s − 8·39-s − 2·41-s + ⋯ |
| L(s) = 1 | − 1.15i·3-s + (0.447 + 0.894i)5-s − 0.333·9-s − 1.20·11-s − 1.10i·13-s + (1.03 − 0.516i)15-s − 0.917·19-s − 0.417i·23-s + (−0.600 + 0.800i)25-s − 0.769i·27-s − 0.371·29-s + 1.39i·33-s − 0.657i·37-s − 1.28·39-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9921045380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9921045380\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 14iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 16iT - 97T^{2} \) |
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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
−8.593897471046692907627939762493, −7.85390848862261974976355825286, −7.32814370447557757601087272345, −6.53884280212563627203727225074, −5.86056772947807781412996216914, −5.02469437549328439557199908410, −3.58485400635582933248857258888, −2.57097691746894201931413493371, −1.93849356072506714350232402441, −0.33352152405547286101464605490,
1.58107359650683069944637109463, 2.75555881058552054109776838878, 4.05010593479822587704589812835, 4.59537567361456782616078085091, 5.29798925345426560110549974082, 6.11495671370011548301406368150, 7.20294575888161820302129955739, 8.235585501317652893810304862043, 8.848426049592365618984359381360, 9.637244616815914765340100375155
