L(s) = 1 | + (1.98 − 0.205i)2-s + (3.91 − 0.817i)4-s + 0.0956·5-s − 9.56i·7-s + (7.62 − 2.43i)8-s + (0.190 − 0.0196i)10-s − 7.63i·11-s + 7.83·13-s + (−1.96 − 19.0i)14-s + (14.6 − 6.40i)16-s − 16.6·17-s + 4.35i·19-s + (0.374 − 0.0781i)20-s + (−1.56 − 15.1i)22-s + 6.37i·23-s + ⋯ |
L(s) = 1 | + (0.994 − 0.102i)2-s + (0.978 − 0.204i)4-s + 0.0191·5-s − 1.36i·7-s + (0.952 − 0.303i)8-s + (0.0190 − 0.00196i)10-s − 0.693i·11-s + 0.603·13-s + (−0.140 − 1.35i)14-s + (0.916 − 0.400i)16-s − 0.982·17-s + 0.229i·19-s + (0.0187 − 0.00390i)20-s + (−0.0712 − 0.690i)22-s + 0.277i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.436562462\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.436562462\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.98 + 0.205i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
good | 5 | \( 1 - 0.0956T + 25T^{2} \) |
| 7 | \( 1 + 9.56iT - 49T^{2} \) |
| 11 | \( 1 + 7.63iT - 121T^{2} \) |
| 13 | \( 1 - 7.83T + 169T^{2} \) |
| 17 | \( 1 + 16.6T + 289T^{2} \) |
| 23 | \( 1 - 6.37iT - 529T^{2} \) |
| 29 | \( 1 + 19.5T + 841T^{2} \) |
| 31 | \( 1 + 57.7iT - 961T^{2} \) |
| 37 | \( 1 - 50.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 57.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 13.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 42.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 40.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 89.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 30.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 61.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 134. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 101.T + 9.40e3T^{2} \) |
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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
−10.36177170100619749616146497772, −9.433346577755402956553415935747, −8.035900181323196350126840754302, −7.35071880340907618902886047654, −6.34025343267424889425684382621, −5.62220042153602299631399884379, −4.15424942305980860958377775996, −3.87765203758918678694807216721, −2.38736736531218908138839442069, −0.888757839297214592414161103109,
1.88202011468809032467808246292, 2.77285934129291287584869357153, 4.06309613144540066633073390656, 5.03183136536494767939895153669, 5.91516247440642483539477128274, 6.64461394792370025955721046679, 7.72198540722366608542546648651, 8.705683585758491812731389721479, 9.551438379418776184936243391278, 10.79066557971216386077099949180