L(s) = 1 | + (1.29 + 0.576i)2-s − i·3-s + (1.33 + 1.48i)4-s + (0.576 − 1.29i)6-s + 1.97·7-s + (0.867 + 2.69i)8-s − 9-s − 1.43i·11-s + (1.48 − 1.33i)12-s − 0.241i·13-s + (2.55 + 1.13i)14-s + (−0.430 + 3.97i)16-s + 7.38·17-s + (−1.29 − 0.576i)18-s + 3.04i·19-s + ⋯ |
L(s) = 1 | + (0.913 + 0.407i)2-s − 0.577i·3-s + (0.667 + 0.744i)4-s + (0.235 − 0.527i)6-s + 0.747·7-s + (0.306 + 0.951i)8-s − 0.333·9-s − 0.431i·11-s + (0.429 − 0.385i)12-s − 0.0669i·13-s + (0.682 + 0.304i)14-s + (−0.107 + 0.994i)16-s + 1.79·17-s + (−0.304 − 0.135i)18-s + 0.697i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.70919 + 0.425837i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.70919 + 0.425837i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.29 - 0.576i)T \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 + 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 0.241iT - 13T^{2} \) |
| 17 | \( 1 - 7.38T + 17T^{2} \) |
| 19 | \( 1 - 3.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.874T + 23T^{2} \) |
| 29 | \( 1 + 9.07iT - 29T^{2} \) |
| 31 | \( 1 + 7.44T + 31T^{2} \) |
| 37 | \( 1 - 8.81iT - 37T^{2} \) |
| 41 | \( 1 + 1.91T + 41T^{2} \) |
| 43 | \( 1 + 11.2iT - 43T^{2} \) |
| 47 | \( 1 - 3.34T + 47T^{2} \) |
| 53 | \( 1 - 9.20iT - 53T^{2} \) |
| 59 | \( 1 + 6.43iT - 59T^{2} \) |
| 61 | \( 1 - 4.57iT - 61T^{2} \) |
| 67 | \( 1 - 4.86iT - 67T^{2} \) |
| 71 | \( 1 + 8.21T + 71T^{2} \) |
| 73 | \( 1 - 4.12T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 12.3iT - 83T^{2} \) |
| 89 | \( 1 + 8.08T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−11.00392052778941069711791334951, −9.969600843357573532879450366896, −8.488553424382952677387906281674, −7.88709990641552762827197711626, −7.17018283706080543878767299439, −5.91955173920481645116687039909, −5.44168921213035579537916499124, −4.13144826009036320025101826925, −3.04679607102503896855666445400, −1.63916164355604424296735014084,
1.54742336794360798497906840171, 3.00509348355200390685138954394, 4.00445723364029021260715701800, 5.05686014890892978787199227387, 5.59580870556383367900069313382, 6.94944112096665243952875224026, 7.84056499707213385738348136824, 9.144084371484516344277340466532, 9.956969796536183313650350395953, 10.82274591911613687981245623746