| L(s) = 1 | + (−0.227 + 2.60i)2-s + (−2.46 − 1.71i)3-s + (−2.79 − 0.492i)4-s + (2.63 − 4.25i)5-s + (5.02 − 6.02i)6-s + (4.66 + 3.26i)7-s + (−0.786 + 2.93i)8-s + (3.11 + 8.44i)9-s + (10.4 + 7.82i)10-s + (12.9 + 4.70i)11-s + (6.03 + 6.00i)12-s + (−0.714 − 8.16i)13-s + (−9.56 + 11.3i)14-s + (−13.7 + 5.94i)15-s + (−18.1 − 6.59i)16-s + (7.23 + 26.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.113 + 1.30i)2-s + (−0.820 − 0.571i)3-s + (−0.698 − 0.123i)4-s + (0.526 − 0.850i)5-s + (0.838 − 1.00i)6-s + (0.665 + 0.466i)7-s + (−0.0982 + 0.366i)8-s + (0.346 + 0.938i)9-s + (1.04 + 0.782i)10-s + (1.17 + 0.427i)11-s + (0.503 + 0.500i)12-s + (−0.0549 − 0.628i)13-s + (−0.683 + 0.814i)14-s + (−0.917 + 0.396i)15-s + (−1.13 − 0.412i)16-s + (0.425 + 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.01311 + 0.780954i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.01311 + 0.780954i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.46 + 1.71i)T \) |
| 5 | \( 1 + (-2.63 + 4.25i)T \) |
| good | 2 | \( 1 + (0.227 - 2.60i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-4.66 - 3.26i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-12.9 - 4.70i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.714 + 8.16i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-7.23 - 26.9i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (-26.2 + 15.1i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (14.8 - 10.3i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (-21.2 - 25.2i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-4.67 + 26.5i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-10.0 - 37.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (0.528 + 0.443i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (3.77 + 8.09i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (32.6 + 22.8i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (29.6 + 29.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (36.4 + 100. i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-10.9 - 62.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (76.4 - 6.68i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (16.7 - 29.0i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-3.38 + 12.6i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (45.0 + 53.7i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-18.4 - 1.61i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (71.9 - 41.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (24.7 - 11.5i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−13.30719774221447527522204908466, −12.18985921480546278386940737959, −11.48252537786614994628806534101, −9.879899786336574772490065771942, −8.585214918936415184006406115072, −7.77865091574543099967157285843, −6.51843368165740313413326308426, −5.65967645295284870994140939424, −4.81740072954573493223488445018, −1.55545904325166220536186760195,
1.25169810771858164307871845870, 3.18373235810020877523028561687, 4.43181969525750588651912475184, 6.06155722783184807850168485964, 7.18586720914989338835218014698, 9.341962818177667723801178249675, 9.913468391934762801003843927044, 10.89551477971610201959170595253, 11.62621038011118222362178358694, 12.10042496371423211655845904052
