L(s) = 1 | + (−2.64 − 1.41i)3-s + 0.913i·5-s − 2.64·7-s + (5 + 7.48i)9-s + 14.5i·11-s + 0.583·13-s + (1.29 − 2.41i)15-s − 21.5i·17-s + 16·19-s + (7.00 + 3.74i)21-s + 38.8i·23-s + 24.1·25-s + (−2.64 − 26.8i)27-s − 35.7i·29-s − 58.4·31-s + ⋯ |
L(s) = 1 | + (−0.881 − 0.471i)3-s + 0.182i·5-s − 0.377·7-s + (0.555 + 0.831i)9-s + 1.32i·11-s + 0.0448·13-s + (0.0861 − 0.161i)15-s − 1.26i·17-s + 0.842·19-s + (0.333 + 0.178i)21-s + 1.68i·23-s + 0.966·25-s + (−0.0979 − 0.995i)27-s − 1.23i·29-s − 1.88·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3185654228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3185654228\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.64 + 1.41i)T \) |
| 7 | \( 1 + 2.64T \) |
good | 5 | \( 1 - 0.913iT - 25T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 0.583T + 169T^{2} \) |
| 17 | \( 1 + 21.5iT - 289T^{2} \) |
| 19 | \( 1 - 16T + 361T^{2} \) |
| 23 | \( 1 - 38.8iT - 529T^{2} \) |
| 29 | \( 1 + 35.7iT - 841T^{2} \) |
| 31 | \( 1 + 58.4T + 961T^{2} \) |
| 37 | \( 1 + 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 8.75iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 58.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 38.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 70.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 17.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 72.3T + 5.32e3T^{2} \) |
| 79 | \( 1 - 20.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 145. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 53.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111.T + 9.40e3T^{2} \) |
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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
−9.641856802518247616112669539516, −9.317416117276987337178840551552, −7.69080028236637264920758587570, −7.31720160020989646235824054314, −6.61757445501993412963975629723, −5.50019875670847872288061058983, −4.97921362458184043534002620827, −3.78747433559934356180451434076, −2.48075371278524351111877364583, −1.32837603725862881117031409016,
0.11389118795495458645052159688, 1.30871582720775857834218977091, 3.10774058690798823714467336966, 3.87973272618361323104634838217, 4.97204016114713753691591037913, 5.75150625485481699378560312337, 6.41277195366614589837326064123, 7.27340468828697238122261237489, 8.583834878654393135891319251510, 8.969191236428461689364035181758