| L(s) = 1 | + (−1.30 − 0.0149i)2-s + (−0.660 + 2.89i)3-s + (−0.302 − 0.00697i)4-s + (0.484 − 0.911i)5-s + (0.903 − 3.75i)6-s + (−1.50 − 3.70i)7-s + (2.99 + 0.103i)8-s + (−5.23 − 2.51i)9-s + (−0.645 + 1.17i)10-s + (4.17 + 4.34i)11-s + (0.220 − 0.871i)12-s + (3.30 + 0.498i)13-s + (1.90 + 4.85i)14-s + (2.31 + 2.00i)15-s + (−3.29 − 0.151i)16-s + (−0.132 + 0.785i)17-s + ⋯ |
| L(s) = 1 | + (−0.921 − 0.0105i)2-s + (−0.381 + 1.67i)3-s + (−0.151 − 0.00348i)4-s + (0.216 − 0.407i)5-s + (0.368 − 1.53i)6-s + (−0.568 − 1.40i)7-s + (1.06 + 0.0366i)8-s + (−1.74 − 0.839i)9-s + (−0.204 + 0.372i)10-s + (1.25 + 1.31i)11-s + (0.0635 − 0.251i)12-s + (0.917 + 0.138i)13-s + (0.509 + 1.29i)14-s + (0.597 + 0.517i)15-s + (−0.824 − 0.0379i)16-s + (−0.0321 + 0.190i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0315 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.483396 + 0.498903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.483396 + 0.498903i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 547 | \( 1 + (0.112 - 23.3i)T \) |
| good | 2 | \( 1 + (1.30 + 0.0149i)T + (1.99 + 0.0460i)T^{2} \) |
| 3 | \( 1 + (0.660 - 2.89i)T + (-2.70 - 1.30i)T^{2} \) |
| 5 | \( 1 + (-0.484 + 0.911i)T + (-2.79 - 4.14i)T^{2} \) |
| 7 | \( 1 + (1.50 + 3.70i)T + (-5.02 + 4.87i)T^{2} \) |
| 11 | \( 1 + (-4.17 - 4.34i)T + (-0.442 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-3.30 - 0.498i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (0.132 - 0.785i)T + (-16.0 - 5.56i)T^{2} \) |
| 19 | \( 1 + (-1.46 + 0.792i)T + (10.4 - 15.8i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 0.310i)T + (20.6 + 10.2i)T^{2} \) |
| 29 | \( 1 + (2.94 + 1.67i)T + (14.7 + 24.9i)T^{2} \) |
| 31 | \( 1 + (4.53 + 1.63i)T + (23.8 + 19.7i)T^{2} \) |
| 37 | \( 1 + (0.0216 - 0.0174i)T + (7.81 - 36.1i)T^{2} \) |
| 41 | \( 1 + (-2.89 - 5.01i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.311 - 3.17i)T + (-42.1 - 8.35i)T^{2} \) |
| 47 | \( 1 + (-13.0 + 1.05i)T + (46.3 - 7.53i)T^{2} \) |
| 53 | \( 1 + (9.31 - 3.22i)T + (41.6 - 32.8i)T^{2} \) |
| 59 | \( 1 + (-1.71 + 0.732i)T + (40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (5.15 - 2.40i)T + (39.1 - 46.8i)T^{2} \) |
| 67 | \( 1 + (-11.3 - 11.0i)T + (1.92 + 66.9i)T^{2} \) |
| 71 | \( 1 + (-13.1 + 2.60i)T + (65.6 - 27.0i)T^{2} \) |
| 73 | \( 1 + (3.39 - 2.93i)T + (10.4 - 72.2i)T^{2} \) |
| 79 | \( 1 + (-2.04 - 1.46i)T + (25.4 + 74.7i)T^{2} \) |
| 83 | \( 1 + (-6.69 - 4.23i)T + (35.5 + 74.9i)T^{2} \) |
| 89 | \( 1 + (-1.06 - 12.2i)T + (-87.6 + 15.2i)T^{2} \) |
| 97 | \( 1 + (-1.88 - 14.1i)T + (-93.6 + 25.3i)T^{2} \) |
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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
−10.76934164949863035390009613634, −9.876927148998895855869249451612, −9.440221562285967329582946850846, −8.959420159649672303593033896629, −7.57630505188129038327379025443, −6.57287079862050717144867992026, −5.17233284912803547339749271404, −4.14890141047848390385002115635, −3.84562936822870608975676260145, −1.14322002142198601230184590832,
0.74765788098291984750568238636, 1.98306507743481207202138693337, 3.38113514045837349032263874890, 5.55656388963720588362402301503, 6.20167225256447596001654456098, 6.93212448131180642119511226811, 8.034999532360177134936100748153, 8.854223403496628423241135641413, 9.208018417604912002289752997384, 10.75764000783324298149923888796
