| L(s) = 1 | + 2.23i·3-s − 2.00·9-s − 2.23i·11-s + 4i·13-s − 3·17-s + 2.23i·19-s − 8.94·23-s + 2.23i·27-s − 4i·29-s + 8.94·31-s + 5.00·33-s − 8i·37-s − 8.94·39-s − 5·41-s + 8.94i·43-s + ⋯ |
| L(s) = 1 | + 1.29i·3-s − 0.666·9-s − 0.674i·11-s + 1.10i·13-s − 0.727·17-s + 0.512i·19-s − 1.86·23-s + 0.430i·27-s − 0.742i·29-s + 1.60·31-s + 0.870·33-s − 1.31i·37-s − 1.43·39-s − 0.780·41-s + 1.36i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4456246182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4456246182\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 + 8.94T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 - 8.94iT - 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 6.70iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.290257268190771437970249935508, −8.449163157436988419708876843877, −7.916137813142783738822390315149, −6.65426973723809780798854128006, −6.12255237798878427844622807957, −5.17370950836085406898054375509, −4.25534330486328723020255699387, −3.97968293576252330932730150091, −2.87023869192447953628780265014, −1.72442223476160947627437877776,
0.13075382116840722195939797025, 1.41604291272217896140180619231, 2.25487976650023687003794463818, 3.16885017314654111631878258783, 4.38403108217024187195938086953, 5.18745158337838236928215089003, 6.25796643600142511885926106246, 6.65509592633679720264167186216, 7.46941404249055981619472336369, 8.136344312040568156017116394619
