L(s) = 1 | + (−1.22 − 1.22i)3-s + (−0.775 + 0.775i)7-s + 2.99i·9-s − 2.89·11-s + (5.87 + 5.87i)13-s + (4.44 − 4.44i)17-s + 0.101i·19-s + 1.89·21-s + (−25.3 − 25.3i)23-s + (3.67 − 3.67i)27-s + 32.2i·29-s + 3.69·31-s + (3.55 + 3.55i)33-s + (42.6 − 42.6i)37-s − 14.3i·39-s + ⋯ |
L(s) = 1 | + (−0.408 − 0.408i)3-s + (−0.110 + 0.110i)7-s + 0.333i·9-s − 0.263·11-s + (0.452 + 0.452i)13-s + (0.261 − 0.261i)17-s + 0.00531i·19-s + 0.0904·21-s + (−1.10 − 1.10i)23-s + (0.136 − 0.136i)27-s + 1.11i·29-s + 0.119·31-s + (0.107 + 0.107i)33-s + (1.15 − 1.15i)37-s − 0.369i·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
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3269783480\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3269783480\) |
\(L(2)\) |
|
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 | \( 1 + (0.775 - 0.775i)T - 49iT^{2} \) |
| 11 | \( 1 + 2.89T + 121T^{2} \) |
| 13 | \( 1 + (-5.87 - 5.87i)T + 169iT^{2} \) |
| 17 | \( 1 + (-4.44 + 4.44i)T - 289iT^{2} \) |
| 19 | \( 1 - 0.101iT - 361T^{2} \) |
| 23 | \( 1 + (25.3 + 25.3i)T + 529iT^{2} \) |
| 29 | \( 1 - 32.2iT - 841T^{2} \) |
| 31 | \( 1 - 3.69T + 961T^{2} \) |
| 37 | \( 1 + (-42.6 + 42.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 12.8T + 1.68e3T^{2} \) |
| 43 | \( 1 + (49.2 + 49.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (2.85 - 2.85i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (13.1 + 13.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 76.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 103.T + 3.72e3T^{2} \) |
| 67 | \( 1 + (47.6 - 47.6i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 29.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-3.50 - 3.50i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 87.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (81.7 + 81.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 96.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-54.2 + 54.2i)T - 9.40e3iT^{2} \) |
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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
−9.112502595730814084580849886729, −8.313278373441961852812056575166, −7.47790906889535580210894405289, −6.59688514291544328488863509801, −5.90321134072117137282059336127, −4.95005816462521775049791746631, −3.94690575462706429459807059093, −2.70379324117270388106961852036, −1.54337969002848016527097969691, −0.10396537190795019219888463188,
1.40969149857387187415047990993, 2.92832272943218759079811889149, 3.89610533643209519376938058670, 4.82130953414174271897361389793, 5.83701680279340757981516828622, 6.37174733711220848196595566408, 7.65031034221632719428924140866, 8.197139313602988835767355521492, 9.317079475996592892680230192552, 10.02874523748906023169657968367