L(s) = 1 | + (−1.99 + 1.14i)2-s + (0.640 − 1.10i)4-s + (0.140 + 0.243i)5-s − 5.24·7-s − 6.24i·8-s + (−0.558 − 0.322i)10-s + 1.15·11-s + (−6.32 − 3.65i)13-s + (10.4 − 6.02i)14-s + (9.74 + 16.8i)16-s + (−7.52 − 13.0i)17-s + (1.52 − 18.9i)19-s + 0.359·20-s + (−2.30 + 1.32i)22-s + (13.3 − 23.1i)23-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.574i)2-s + (0.160 − 0.277i)4-s + (0.0280 + 0.0486i)5-s − 0.748·7-s − 0.781i·8-s + (−0.0558 − 0.0322i)10-s + 0.105·11-s + (−0.486 − 0.280i)13-s + (0.745 − 0.430i)14-s + (0.608 + 1.05i)16-s + (−0.442 − 0.766i)17-s + (0.0801 − 0.996i)19-s + 0.0179·20-s + (−0.104 + 0.0604i)22-s + (0.581 − 1.00i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.355385 - 0.249789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.355385 - 0.249789i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 + (-1.52 + 18.9i)T \) |
good | 2 | \( 1 + (1.99 - 1.14i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 + (-0.140 - 0.243i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + 5.24T + 49T^{2} \) |
| 11 | \( 1 - 1.15T + 121T^{2} \) |
| 13 | \( 1 + (6.32 + 3.65i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (7.52 + 13.0i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-13.3 + 23.1i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-7.76 - 4.48i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + 11.7iT - 961T^{2} \) |
| 37 | \( 1 + 36.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (40.3 - 23.3i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-1.64 - 2.84i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (26.3 - 45.5i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (88.0 + 50.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-39.8 + 22.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (50.7 - 87.8i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (50.9 + 29.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-90.7 + 52.3i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-63.1 - 109. i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (55.9 - 32.2i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 22.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + (107. + 62.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-162. + 93.9i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−12.48604514729001346255055800482, −11.09232908583042051294564528856, −9.962549424525084659648047355857, −9.242806013623952214345788438984, −8.305964954437610191710009227908, −7.10262706441250010810827399817, −6.43212382070534489431598265091, −4.67333057434990471743608274193, −2.94174367412872288221104885647, −0.36986254549519978206089716352,
1.62612383278772591596437753402, 3.31807540632195250449342338073, 5.13355208499972933254737860788, 6.51828991937071483147868387635, 7.84798964377039225085680674064, 8.940039797321472243615854969932, 9.715196197719328754295403823573, 10.51609081957318441752515149050, 11.52653171695119376463640292794, 12.49455378629159046925343471995