L(s) = 1 | + (3.22 + 3.22i)7-s + 6.89·11-s + (18.1 − 18.1i)13-s + (0.449 + 0.449i)17-s + 9.89i·19-s + (−10.6 + 10.6i)23-s − 36.2i·29-s + 25.6·31-s + (13.3 + 13.3i)37-s + 3.10·41-s + (2.72 − 2.72i)43-s + (−37.1 − 37.1i)47-s − 28.2i·49-s + (−65.1 + 65.1i)53-s − 80.3i·59-s + ⋯ |
L(s) = 1 | + (0.460 + 0.460i)7-s + 0.627·11-s + (1.39 − 1.39i)13-s + (0.0264 + 0.0264i)17-s + 0.520i·19-s + (−0.463 + 0.463i)23-s − 1.25i·29-s + 0.828·31-s + (0.359 + 0.359i)37-s + 0.0756·41-s + (0.0634 − 0.0634i)43-s + (−0.790 − 0.790i)47-s − 0.575i·49-s + (−1.23 + 1.23i)53-s − 1.36i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.424156263\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.424156263\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 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
−8.933010280442623421715271936595, −8.175565454633090524205308396847, −7.78036160891436899135123233469, −6.40523666600902437045462839942, −5.94357818871456340610994108470, −5.04714981209664005778255518086, −3.95245496396433092722774810756, −3.17119823387628451746276965444, −1.90191784743471402843478930757, −0.802875464227899357194357987825,
1.01684206206432403194574639839, 1.90833542025334258861219376472, 3.33172862465264747248774571570, 4.20390811108780279928488964184, 4.85020118695578663955095984543, 6.18602089986295702647879889498, 6.61106695783776960435571379272, 7.54607199469190833265416257923, 8.483226951548077228107296496924, 9.024975844810148351456435429886