| L(s) = 1 | + (0.272 − 3.11i)2-s + (−2.25 − 1.98i)3-s + (−5.69 − 1.00i)4-s + (−2.39 − 4.38i)5-s + (−6.78 + 6.48i)6-s + (0.693 + 0.485i)7-s + (−1.44 + 5.37i)8-s + (1.15 + 8.92i)9-s + (−14.3 + 6.25i)10-s + (9.25 + 3.36i)11-s + (10.8 + 13.5i)12-s + (−0.242 − 2.77i)13-s + (1.70 − 2.02i)14-s + (−3.30 + 14.6i)15-s + (−5.37 − 1.95i)16-s + (−4.24 − 15.8i)17-s + ⋯ |
| L(s) = 1 | + (0.136 − 1.55i)2-s + (−0.751 − 0.660i)3-s + (−1.42 − 0.250i)4-s + (−0.478 − 0.877i)5-s + (−1.13 + 1.08i)6-s + (0.0990 + 0.0693i)7-s + (−0.180 + 0.671i)8-s + (0.128 + 0.991i)9-s + (−1.43 + 0.625i)10-s + (0.841 + 0.306i)11-s + (0.903 + 1.12i)12-s + (−0.0186 − 0.213i)13-s + (0.121 − 0.144i)14-s + (−0.220 + 0.975i)15-s + (−0.335 − 0.122i)16-s + (−0.249 − 0.931i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.442 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.429175 + 0.690553i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.429175 + 0.690553i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.25 + 1.98i)T \) |
| 5 | \( 1 + (2.39 + 4.38i)T \) |
| good | 2 | \( 1 + (-0.272 + 3.11i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (-0.693 - 0.485i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (-9.25 - 3.36i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (0.242 + 2.77i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (4.24 + 15.8i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (22.9 - 13.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-15.4 + 10.8i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (3.76 + 4.49i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (-7.87 + 44.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-6.47 - 24.1i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (12.8 + 10.7i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (27.0 + 57.9i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (20.9 + 14.6i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (-13.5 - 13.5i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (2.57 + 7.08i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-19.2 - 109. i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-110. + 9.70i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (-68.3 + 118. i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-6.89 + 25.7i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-88.3 - 105. i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-75.5 - 6.60i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-63.7 + 36.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-48.9 + 22.8i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−12.05680843029527992189084588711, −11.64454031220012541646643825438, −10.61724965027796789051077226396, −9.443763131349225895899761543118, −8.261437449989945344401933535197, −6.76226358723724335941946831773, −5.08942733393860046391641213911, −4.03587198055383530887033246897, −2.02510758568338902349514158045, −0.58589354062254209663824850525,
3.79569913191723272973153145253, 4.92322681662146115599649605204, 6.40941829898818770651012750896, 6.71306318939741517795339255991, 8.198159432003791433533168556572, 9.281759342211928462036216184307, 10.75752257230783334078144576752, 11.40559366531687222083221328796, 12.82738760017298757954958963869, 14.28584971343436812586088655090
