| L(s) = 1 | + (−1.25 − 0.653i)2-s + (0.0781 + 3.18i)3-s + (1.14 + 1.63i)4-s + (−0.0822 − 0.297i)5-s + (1.98 − 4.04i)6-s + (0.751 − 1.58i)7-s + (−0.365 − 2.80i)8-s + (−7.12 + 0.349i)9-s + (−0.0910 + 0.426i)10-s + (−3.37 − 2.37i)11-s + (−5.12 + 3.77i)12-s + (−4.35 + 0.537i)13-s + (−1.98 + 1.50i)14-s + (0.938 − 0.284i)15-s + (−1.37 + 3.75i)16-s + (−0.773 + 2.55i)17-s + ⋯ |
| L(s) = 1 | + (−0.886 − 0.462i)2-s + (0.0451 + 1.83i)3-s + (0.572 + 0.819i)4-s + (−0.0367 − 0.132i)5-s + (0.809 − 1.65i)6-s + (0.284 − 0.600i)7-s + (−0.129 − 0.991i)8-s + (−2.37 + 0.116i)9-s + (−0.0287 + 0.134i)10-s + (−1.01 − 0.716i)11-s + (−1.48 + 1.08i)12-s + (−1.20 + 0.149i)13-s + (−0.529 + 0.401i)14-s + (0.242 − 0.0735i)15-s + (−0.343 + 0.939i)16-s + (−0.187 + 0.618i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0259304 - 0.0715593i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0259304 - 0.0715593i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.25 + 0.653i)T \) |
| good | 3 | \( 1 + (-0.0781 - 3.18i)T + (-2.99 + 0.147i)T^{2} \) |
| 5 | \( 1 + (0.0822 + 0.297i)T + (-4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.751 + 1.58i)T + (-4.44 - 5.41i)T^{2} \) |
| 11 | \( 1 + (3.37 + 2.37i)T + (3.70 + 10.3i)T^{2} \) |
| 13 | \( 1 + (4.35 - 0.537i)T + (12.6 - 3.15i)T^{2} \) |
| 17 | \( 1 + (0.773 - 2.55i)T + (-14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (-0.456 + 6.19i)T + (-18.7 - 2.78i)T^{2} \) |
| 23 | \( 1 + (6.74 - 5.00i)T + (6.67 - 22.0i)T^{2} \) |
| 29 | \( 1 + (10.3 - 2.32i)T + (26.2 - 12.3i)T^{2} \) |
| 31 | \( 1 + (-1.03 - 1.54i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-2.04 - 4.07i)T + (-22.0 + 29.7i)T^{2} \) |
| 41 | \( 1 + (-2.56 - 1.53i)T + (19.3 + 36.1i)T^{2} \) |
| 43 | \( 1 + (8.30 + 8.72i)T + (-2.10 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-3.25 - 2.66i)T + (9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-1.89 - 0.425i)T + (47.9 + 22.6i)T^{2} \) |
| 59 | \( 1 + (-0.455 + 3.69i)T + (-57.2 - 14.3i)T^{2} \) |
| 61 | \( 1 + (-1.47 - 3.33i)T + (-40.9 + 45.1i)T^{2} \) |
| 67 | \( 1 + (2.88 - 7.48i)T + (-49.6 - 44.9i)T^{2} \) |
| 71 | \( 1 + (0.990 + 1.09i)T + (-6.95 + 70.6i)T^{2} \) |
| 73 | \( 1 + (-5.22 - 11.0i)T + (-46.3 + 56.4i)T^{2} \) |
| 79 | \( 1 + (0.267 - 2.71i)T + (-77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (-3.29 + 6.55i)T + (-49.4 - 66.6i)T^{2} \) |
| 89 | \( 1 + (13.8 + 10.2i)T + (25.8 + 85.1i)T^{2} \) |
| 97 | \( 1 + (0.974 + 0.193i)T + (89.6 + 37.1i)T^{2} \) |
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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
−11.07551210148032181139946640956, −10.41450173117616581626900401137, −9.830448444374070449913274283454, −9.011191616727963693136509910085, −8.232378399584491291460386448736, −7.24594434796564428111390367744, −5.60207197691074574072865517983, −4.59428102722166154998327488006, −3.63180849885956804490573114976, −2.55508780109026826665288462011,
0.05390489917436322541098433952, 1.93063321596333338652798471685, 2.50449535631562134031056333360, 5.21686689487635342827501835060, 5.98762135880266783649898267730, 7.01014919267934294163158469638, 7.75153435854653494360209149607, 8.069848314739795363117343691197, 9.270308815288057788485930744867, 10.23387158858141201811942595341
