| L(s) = 1 | + (−2.37 + 0.866i)2-s + (3.37 − 2.83i)4-s + (0.0812 + 0.460i)5-s + (−2.47 − 2.07i)7-s + (−3.05 + 5.28i)8-s + (−0.592 − 1.02i)10-s + (−0.539 + 3.05i)11-s + (2.05 + 0.747i)13-s + (7.67 + 2.79i)14-s + (1.15 − 6.53i)16-s + (−1.5 − 2.59i)17-s + (−0.0209 + 0.0362i)19-s + (1.58 + 1.32i)20-s + (−1.36 − 7.74i)22-s + (4.67 − 3.92i)23-s + ⋯ |
| L(s) = 1 | + (−1.68 + 0.612i)2-s + (1.68 − 1.41i)4-s + (0.0363 + 0.206i)5-s + (−0.934 − 0.783i)7-s + (−1.07 + 1.86i)8-s + (−0.187 − 0.324i)10-s + (−0.162 + 0.922i)11-s + (0.569 + 0.207i)13-s + (2.05 + 0.746i)14-s + (0.288 − 1.63i)16-s + (−0.363 − 0.630i)17-s + (−0.00480 + 0.00832i)19-s + (0.353 + 0.296i)20-s + (−0.291 − 1.65i)22-s + (0.975 − 0.818i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0581 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.185860 - 0.197000i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.185860 - 0.197000i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.37 - 0.866i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.0812 - 0.460i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (2.47 + 2.07i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.539 - 3.05i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-2.05 - 0.747i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.0209 - 0.0362i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.67 + 3.92i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (6.17 - 2.24i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (4.76 - 4.00i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (1.79 + 3.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.23 + 2.63i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.102 - 0.579i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (7.40 + 6.20i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 4.95T + 53T^{2} \) |
| 59 | \( 1 + (1.48 + 8.40i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (0.971 + 0.815i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (9.40 + 3.42i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.91 + 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.11 + 7.13i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (10.3 - 3.77i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-1.41 + 0.516i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-7.93 + 13.7i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.23 + 18.3i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.10810178173060739706104918166, −9.171354038613052615186793528558, −8.687844077203152640663428598585, −7.35708462378287347570092244614, −7.02195835893292388250335393097, −6.31996426772637409444374699483, −4.93282899135714454151038459976, −3.33975228418638517137297466404, −1.82140077763391736019569406286, −0.25249697471821080337441353075,
1.35313326382164775267940421347, 2.75020175818702484396726527528, 3.54457905232405511952809491081, 5.53396240141280029888023670253, 6.44284519321611044766302346366, 7.45357357108776627212413341619, 8.463513422206343942328544660947, 8.968629371050873365898862125325, 9.578726260124383193940488252280, 10.53062875453819822716505715867
