| L(s) = 1 | + (−1.99 − 0.726i)2-s + (1.92 + 1.61i)4-s + (0.359 − 2.03i)5-s + (3.71 − 3.11i)7-s + (−0.547 − 0.949i)8-s + (−2.20 + 3.81i)10-s + (−0.720 − 4.08i)11-s + (1.14 − 0.415i)13-s + (−9.68 + 3.52i)14-s + (−0.469 − 2.66i)16-s + (1.18 − 2.04i)17-s + (0.919 + 1.59i)19-s + (3.99 − 3.34i)20-s + (−1.53 + 8.68i)22-s + (3.29 + 2.76i)23-s + ⋯ |
| L(s) = 1 | + (−1.41 − 0.513i)2-s + (0.963 + 0.808i)4-s + (0.160 − 0.912i)5-s + (1.40 − 1.17i)7-s + (−0.193 − 0.335i)8-s + (−0.695 + 1.20i)10-s + (−0.217 − 1.23i)11-s + (0.316 − 0.115i)13-s + (−2.58 + 0.941i)14-s + (−0.117 − 0.665i)16-s + (0.286 − 0.496i)17-s + (0.210 + 0.365i)19-s + (0.892 − 0.748i)20-s + (−0.326 + 1.85i)22-s + (0.687 + 0.576i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.403674 - 0.803782i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.403674 - 0.803782i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (1.99 + 0.726i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.359 + 2.03i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.71 + 3.11i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.720 + 4.08i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 0.415i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.18 + 2.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.29 - 2.76i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.80 - 1.01i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 0.947i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (4.48 - 7.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.12 + 0.773i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.952 + 5.40i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (5.50 - 4.61i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 6.32T + 53T^{2} \) |
| 59 | \( 1 + (0.0455 - 0.258i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.41 - 2.86i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.88 - 1.41i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 2.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.38 + 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.27 + 1.55i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (7.94 + 2.89i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-8.48 - 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.887 + 5.03i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.18666147015616875492751585511, −9.105407315039499176224497416799, −8.440738640960861496531787613249, −7.939110142393985828852388521194, −7.05901378864879168463242014099, −5.43407607821265678077940808982, −4.66918509926147703812682351686, −3.20792301084338657274745262102, −1.47071709179171851696291556603, −0.870426376688650794305197492475,
1.61657102127382554144289596392, 2.58419635992268454445859372086, 4.47740701278347380431746483790, 5.59019802494969078708767839880, 6.66506125902201740836914952582, 7.37491021410942897152087841957, 8.229873369143875005231612307600, 8.827385501212033852625896074407, 9.721237974720832975408356035359, 10.54918239999547335716984767909
