| L(s) = 1 | + (−1.58 + 1.21i)2-s + (0.418 − 2.97i)3-s + (1.02 − 3.86i)4-s + (−4.40 + 2.54i)5-s + (2.96 + 5.21i)6-s + (−10.9 − 6.32i)7-s + (3.09 + 7.37i)8-s + (−8.65 − 2.48i)9-s + (3.88 − 9.40i)10-s + (4.51 − 7.81i)11-s + (−11.0 − 4.66i)12-s + (9.68 − 5.59i)13-s + (25.0 − 3.33i)14-s + (5.71 + 14.1i)15-s + (−13.9 − 7.92i)16-s − 19.2·17-s + ⋯ |
| L(s) = 1 | + (−0.792 + 0.609i)2-s + (0.139 − 0.990i)3-s + (0.256 − 0.966i)4-s + (−0.881 + 0.508i)5-s + (0.493 + 0.869i)6-s + (−1.56 − 0.904i)7-s + (0.386 + 0.922i)8-s + (−0.961 − 0.276i)9-s + (0.388 − 0.940i)10-s + (0.410 − 0.710i)11-s + (−0.921 − 0.388i)12-s + (0.744 − 0.430i)13-s + (1.79 − 0.238i)14-s + (0.381 + 0.943i)15-s + (−0.868 − 0.495i)16-s − 1.13·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.201871 - 0.374027i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.201871 - 0.374027i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.58 - 1.21i)T \) |
| 3 | \( 1 + (-0.418 + 2.97i)T \) |
| good | 5 | \( 1 + (4.40 - 2.54i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (10.9 + 6.32i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-4.51 + 7.81i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-9.68 + 5.59i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 19.2T + 289T^{2} \) |
| 19 | \( 1 - 14.2T + 361T^{2} \) |
| 23 | \( 1 + (-4.28 + 2.47i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-7.55 - 4.36i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-33.9 + 19.6i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 19.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (17.3 + 30.1i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (3.02 - 5.24i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (52.8 + 30.4i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 6.53iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (25.0 + 43.4i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-0.149 - 0.0862i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (40.4 + 70.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 50.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 24.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-2.84 - 1.64i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-18.4 + 32.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 13.0T + 7.92e3T^{2} \) |
| 97 | \( 1 + (-44.7 + 77.5i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.93121728385466103272044722907, −13.22248681804098927821552069173, −11.61605215316060403820401768117, −10.63655737139085072701322323289, −9.187249754061590344308782460466, −7.967821227848010172140863125430, −6.91079015922249646181426074153, −6.24433170493665322064412960494, −3.35371173982637205890326184589, −0.45959202735315286864392020706,
3.04537191603836472850918840647, 4.32600000262355891894713105077, 6.56497632376416761054893753378, 8.418138958909289707113843514962, 9.201505022247659180897607458392, 10.01086358351220331399864233329, 11.47040511406488719155681963390, 12.19198036295849159476248971622, 13.38322811074325027350088208249, 15.39469409284268800521518732475
