L(s) = 1 | + (−1.22 − 1.22i)3-s + (3.22 − 3.22i)7-s + 2.99i·9-s + 6.89·11-s + (−18.1 − 18.1i)13-s + (0.449 − 0.449i)17-s + 9.89i·19-s − 7.89·21-s + (10.6 + 10.6i)23-s + (3.67 − 3.67i)27-s − 36.2i·29-s − 25.6·31-s + (−8.44 − 8.44i)33-s + (−13.3 + 13.3i)37-s + 44.3i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (0.460 − 0.460i)7-s + 0.333i·9-s + 0.627·11-s + (−1.39 − 1.39i)13-s + (0.0264 − 0.0264i)17-s + 0.520i·19-s − 0.376·21-s + (0.463 + 0.463i)23-s + (0.136 − 0.136i)27-s − 1.25i·29-s − 0.828·31-s + (−0.256 − 0.256i)33-s + (−0.359 + 0.359i)37-s + 1.13i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6911805764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6911805764\) |
\(L(2)\) |
|
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 | \( 1 + (-3.22 + 3.22i)T - 49iT^{2} \) |
| 11 | \( 1 - 6.89T + 121T^{2} \) |
| 13 | \( 1 + (18.1 + 18.1i)T + 169iT^{2} \) |
| 17 | \( 1 + (-0.449 + 0.449i)T - 289iT^{2} \) |
| 19 | \( 1 - 9.89iT - 361T^{2} \) |
| 23 | \( 1 + (-10.6 - 10.6i)T + 529iT^{2} \) |
| 29 | \( 1 + 36.2iT - 841T^{2} \) |
| 31 | \( 1 + 25.6T + 961T^{2} \) |
| 37 | \( 1 + (13.3 - 13.3i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 3.10T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-2.72 - 2.72i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-37.1 + 37.1i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (65.1 + 65.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 - 80.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-84.3 + 84.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 98.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (52.4 + 52.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 68.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (89.7 + 89.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + 40.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (105. - 105. i)T - 9.40e3iT^{2} \) |
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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
−9.256302603627853728136925075088, −8.050105188752719213245358923247, −7.58668016544304368819683861608, −6.76703533576031316866436478775, −5.69706901611762361662816368027, −5.02203822866499649081291189574, −3.95572689598292033223718867393, −2.72975306040751301312525925244, −1.45996661682411796999710969922, −0.21621820863894672656869992577,
1.56058959668737045327447657314, 2.72060680427003217296404286788, 4.09162910649357422676090907408, 4.81195236622030747295706647165, 5.58418437143446055864585788887, 6.77536323902979271704344716427, 7.22180767340855004450963767374, 8.561448925073863886016987618808, 9.209293925899950203112718084537, 9.783549473760532690889925349783