| L(s) = 1 | + (−1.66 − 1.11i)2-s + (−0.841 + 3.14i)3-s + (1.52 + 3.69i)4-s + (0.797 − 0.213i)5-s + (4.89 − 4.28i)6-s + (−6.95 + 0.832i)7-s + (1.57 − 7.84i)8-s + (−1.36 − 0.786i)9-s + (−1.56 − 0.531i)10-s + (−17.9 − 4.80i)11-s + (−12.8 + 1.68i)12-s + (−5.91 + 5.91i)13-s + (12.4 + 6.34i)14-s + 2.68i·15-s + (−11.3 + 11.2i)16-s + (3.83 + 6.65i)17-s + ⋯ |
| L(s) = 1 | + (−0.831 − 0.556i)2-s + (−0.280 + 1.04i)3-s + (0.381 + 0.924i)4-s + (0.159 − 0.0427i)5-s + (0.815 − 0.714i)6-s + (−0.992 + 0.118i)7-s + (0.196 − 0.980i)8-s + (−0.151 − 0.0873i)9-s + (−0.156 − 0.0531i)10-s + (−1.63 − 0.436i)11-s + (−1.07 + 0.140i)12-s + (−0.455 + 0.455i)13-s + (0.891 + 0.453i)14-s + 0.178i·15-s + (−0.708 + 0.705i)16-s + (0.225 + 0.391i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.889 - 0.456i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0750039 + 0.310279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0750039 + 0.310279i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.66 + 1.11i)T \) |
| 7 | \( 1 + (6.95 - 0.832i)T \) |
| good | 3 | \( 1 + (0.841 - 3.14i)T + (-7.79 - 4.5i)T^{2} \) |
| 5 | \( 1 + (-0.797 + 0.213i)T + (21.6 - 12.5i)T^{2} \) |
| 11 | \( 1 + (17.9 + 4.80i)T + (104. + 60.5i)T^{2} \) |
| 13 | \( 1 + (5.91 - 5.91i)T - 169iT^{2} \) |
| 17 | \( 1 + (-3.83 - 6.65i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (12.0 - 3.22i)T + (312. - 180.5i)T^{2} \) |
| 23 | \( 1 + (0.862 - 1.49i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-18.0 + 18.0i)T - 841iT^{2} \) |
| 31 | \( 1 + (-24.4 + 14.1i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (8.78 - 2.35i)T + (1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 + 22.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (37.8 - 37.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-79.3 - 45.8i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (15.1 - 56.6i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-89.1 - 23.8i)T + (3.01e3 + 1.74e3i)T^{2} \) |
| 61 | \( 1 + (-6.84 - 25.5i)T + (-3.22e3 + 1.86e3i)T^{2} \) |
| 67 | \( 1 + (28.8 - 107. i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 0.687T + 5.04e3T^{2} \) |
| 73 | \( 1 + (31.3 - 18.0i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (17.3 + 9.99i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (95.7 + 95.7i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (66.8 + 38.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 31.0T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.48280782906801942960860696824, −12.66237140662841722556022281136, −11.42268118354353793540261038324, −10.21368327790050329919613338222, −10.03528084980842789283140028478, −8.771554280295514265907737988117, −7.49141621477553860176927430030, −5.85843946865551756590497724202, −4.19059972052017226774230966188, −2.70255973278196155145353523746,
0.28866664594846415393039215622, 2.40626237886601144351081303149, 5.30094227354727625316676318649, 6.52439510357044288095162093110, 7.31481884750760587589919272126, 8.304125476493415919503563544149, 9.869978577209249716618946512123, 10.44038395968579073942920189904, 12.05208461521110558386886149577, 12.95643388250554975166551053588
