L(s) = 1 | + (1.22 + 1.22i)3-s + (−2.44 + 2.44i)7-s + 2.99i·9-s − 6·11-s + (−12.2 − 12.2i)13-s + (14.6 − 14.6i)17-s − 10i·19-s − 5.99·21-s + (29.3 + 29.3i)23-s + (−3.67 + 3.67i)27-s − 48i·29-s + 26·31-s + (−7.34 − 7.34i)33-s + (31.8 − 31.8i)37-s − 29.9i·39-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.349 + 0.349i)7-s + 0.333i·9-s − 0.545·11-s + (−0.942 − 0.942i)13-s + (0.864 − 0.864i)17-s − 0.526i·19-s − 0.285·21-s + (1.27 + 1.27i)23-s + (−0.136 + 0.136i)27-s − 1.65i·29-s + 0.838·31-s + (−0.222 − 0.222i)33-s + (0.860 − 0.860i)37-s − 0.769i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.951144909\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.951144909\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 - 1.22i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (2.44 - 2.44i)T - 49iT^{2} \) |
| 11 | \( 1 + 6T + 121T^{2} \) |
| 13 | \( 1 + (12.2 + 12.2i)T + 169iT^{2} \) |
| 17 | \( 1 + (-14.6 + 14.6i)T - 289iT^{2} \) |
| 19 | \( 1 + 10iT - 361T^{2} \) |
| 23 | \( 1 + (-29.3 - 29.3i)T + 529iT^{2} \) |
| 29 | \( 1 + 48iT - 841T^{2} \) |
| 31 | \( 1 - 26T + 961T^{2} \) |
| 37 | \( 1 + (-31.8 + 31.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 30T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-29.3 - 29.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-14.6 + 14.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-14.6 - 14.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 78iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 2T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-63.6 + 63.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 120T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-83.2 - 83.2i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 74iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (44.0 + 44.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 150iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-4.89 + 4.89i)T - 9.40e3iT^{2} \) |
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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
−9.610849406903405343704872369470, −8.877579288370906368700303526427, −7.61491926869787530694209525554, −7.50722247593958588187665253590, −5.96511496822532849812780138200, −5.27388554708126221264501896748, −4.37895707179509980861773323512, −3.01160205768175695465272359305, −2.60014639425672334863075111385, −0.68144418631172377052370597822,
0.988993051327455225914769412872, 2.30653915746257481772251066723, 3.25907834594960316842357711469, 4.35477384833034909998548394738, 5.32050471397772106590872607226, 6.49994079638653908731622960205, 7.07462301672439647595824734337, 7.957590933246055414987172499425, 8.696721975137459027728589544403, 9.597581108111002216976311124686