| L(s) = 1 | + (−0.832 + 1.81i)2-s + (0.216 − 2.99i)3-s + (−2.61 − 3.02i)4-s + (0.714 + 4.05i)5-s + (5.26 + 2.88i)6-s + (5.27 + 4.42i)7-s + (7.68 − 2.23i)8-s + (−8.90 − 1.29i)9-s + (−7.96 − 2.07i)10-s + (3.61 − 20.5i)11-s + (−9.62 + 7.16i)12-s + (−3.82 + 10.5i)13-s + (−12.4 + 5.90i)14-s + (12.2 − 1.26i)15-s + (−2.33 + 15.8i)16-s + (17.5 − 10.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.416 + 0.909i)2-s + (0.0722 − 0.997i)3-s + (−0.653 − 0.756i)4-s + (0.142 + 0.810i)5-s + (0.876 + 0.480i)6-s + (0.753 + 0.632i)7-s + (0.960 − 0.279i)8-s + (−0.989 − 0.144i)9-s + (−0.796 − 0.207i)10-s + (0.328 − 1.86i)11-s + (−0.802 + 0.597i)12-s + (−0.294 + 0.808i)13-s + (−0.888 + 0.421i)14-s + (0.818 − 0.0840i)15-s + (−0.145 + 0.989i)16-s + (1.03 − 0.595i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33394 + 0.240665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33394 + 0.240665i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.832 - 1.81i)T \) |
| 3 | \( 1 + (-0.216 + 2.99i)T \) |
| good | 5 | \( 1 + (-0.714 - 4.05i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-5.27 - 4.42i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-3.61 + 20.5i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.82 - 10.5i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-17.5 + 10.1i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-25.0 - 14.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-10.6 - 12.6i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-40.4 + 14.7i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-25.1 + 21.1i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (37.0 - 21.3i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-2.39 + 6.57i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (28.3 + 4.99i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-24.4 + 29.1i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 91.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-3.40 - 19.2i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-22.8 + 27.1i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (38.4 - 105. i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (74.4 - 42.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (12.0 - 20.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (25.6 - 9.33i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-16.7 + 6.09i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (81.2 + 46.9i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (3.52 - 19.9i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.88630018264791906729137311888, −11.40236193189144126231043333758, −10.01780227114467261115412349066, −8.774239244182254664778668060349, −8.090375253321089704217614652874, −7.09391177295571365805937099590, −6.13237889793257455766489025430, −5.32447543083405862119819380550, −3.07112602798544766988544310414, −1.18271338668406195688808983433,
1.26096104136936209326270724357, 3.12007930071393595347922670951, 4.70650435312171296784674274878, 4.89533053991030232474452473101, 7.38969364220065430425344445665, 8.370060968094692566081748657355, 9.381032013369288608326206183135, 10.11108996252809665133009585033, 10.79063972517440990556853376535, 12.07323569942093519256222630776
