| L(s) = 1 | + (−1.67 + 2.58i)2-s + (1.40 + 2.65i)3-s + (−2.23 − 5.00i)4-s + (−2.30 − 4.43i)5-s + (−9.20 − 0.821i)6-s + (−6.93 − 1.85i)7-s + (4.51 + 0.714i)8-s + (−5.05 + 7.44i)9-s + (15.3 + 1.49i)10-s + (16.9 − 3.61i)11-s + (10.1 − 12.9i)12-s + (−10.2 − 15.8i)13-s + (16.4 − 14.7i)14-s + (8.53 − 12.3i)15-s + (5.26 − 5.84i)16-s + (−1.65 + 10.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.838 + 1.29i)2-s + (0.467 + 0.883i)3-s + (−0.557 − 1.25i)4-s + (−0.460 − 0.887i)5-s + (−1.53 − 0.136i)6-s + (−0.990 − 0.265i)7-s + (0.564 + 0.0893i)8-s + (−0.562 + 0.827i)9-s + (1.53 + 0.149i)10-s + (1.54 − 0.328i)11-s + (0.845 − 1.07i)12-s + (−0.789 − 1.21i)13-s + (1.17 − 1.05i)14-s + (0.568 − 0.822i)15-s + (0.328 − 0.365i)16-s + (−0.0973 + 0.614i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.933 + 0.357i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.466053 - 0.0862066i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.466053 - 0.0862066i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.40 - 2.65i)T \) |
| 5 | \( 1 + (2.30 + 4.43i)T \) |
| good | 2 | \( 1 + (1.67 - 2.58i)T + (-1.62 - 3.65i)T^{2} \) |
| 7 | \( 1 + (6.93 + 1.85i)T + (42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (-16.9 + 3.61i)T + (110. - 49.2i)T^{2} \) |
| 13 | \( 1 + (10.2 + 15.8i)T + (-68.7 + 154. i)T^{2} \) |
| 17 | \( 1 + (1.65 - 10.4i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (6.87 + 9.46i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-1.29 + 24.6i)T + (-526. - 55.2i)T^{2} \) |
| 29 | \( 1 + (-34.4 - 3.62i)T + (822. + 174. i)T^{2} \) |
| 31 | \( 1 + (2.35 + 22.4i)T + (-939. + 199. i)T^{2} \) |
| 37 | \( 1 + (49.7 + 25.3i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (22.3 + 4.75i)T + (1.53e3 + 683. i)T^{2} \) |
| 43 | \( 1 + (5.63 - 21.0i)T + (-1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (20.7 + 25.5i)T + (-459. + 2.16e3i)T^{2} \) |
| 53 | \( 1 + (1.77 + 11.2i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (9.99 - 47.0i)T + (-3.18e3 - 1.41e3i)T^{2} \) |
| 61 | \( 1 + (-8.82 + 1.87i)T + (3.39e3 - 1.51e3i)T^{2} \) |
| 67 | \( 1 + (-46.7 - 37.8i)T + (933. + 4.39e3i)T^{2} \) |
| 71 | \( 1 + (91.9 + 66.8i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-114. + 58.5i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (111. + 11.7i)T + (6.10e3 + 1.29e3i)T^{2} \) |
| 83 | \( 1 + (37.9 + 98.7i)T + (-5.11e3 + 4.60e3i)T^{2} \) |
| 89 | \( 1 + (-12.0 - 3.92i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (116. + 143. i)T + (-1.95e3 + 9.20e3i)T^{2} \) |
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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
−12.00293589230068097278983469432, −10.44113901126039231175465735007, −9.646880238792750949155954457570, −8.800936799147658496795003113792, −8.294191674197206229807458886281, −7.04665905748432813820200291016, −5.95800252135266221017954700994, −4.68175385259305451093213056055, −3.40221530356433418062768150008, −0.32144539806677612037068812410,
1.64967414791990459056940270837, 2.87939720482972505522998746046, 3.79828260968311812105042570500, 6.51747973264292838768642708181, 7.01303988994379367189908109389, 8.443468349482201996842836468412, 9.384419369441682896392087764385, 9.919724781144139954314946018719, 11.37259421102948263955474307003, 11.98530953475794379525778007080
