| L(s) = 1 | + (0.143 + 1.40i)2-s + (0.584 + 1.63i)3-s + (−1.95 + 0.402i)4-s + (2.25 + 0.819i)5-s + (−2.21 + 1.05i)6-s + (4.05 + 0.715i)7-s + (−0.847 − 2.69i)8-s + (−2.31 + 1.90i)9-s + (−0.830 + 3.28i)10-s + (−1.59 − 4.39i)11-s + (−1.80 − 2.95i)12-s + (−2.03 − 2.42i)13-s + (−0.425 + 5.81i)14-s + (−0.0206 + 4.14i)15-s + (3.67 − 1.57i)16-s + (−5.30 + 3.06i)17-s + ⋯ |
| L(s) = 1 | + (0.101 + 0.994i)2-s + (0.337 + 0.941i)3-s + (−0.979 + 0.201i)4-s + (1.00 + 0.366i)5-s + (−0.902 + 0.430i)6-s + (1.53 + 0.270i)7-s + (−0.299 − 0.954i)8-s + (−0.772 + 0.635i)9-s + (−0.262 + 1.03i)10-s + (−0.482 − 1.32i)11-s + (−0.520 − 0.854i)12-s + (−0.564 − 0.672i)13-s + (−0.113 + 1.55i)14-s + (−0.00533 + 1.07i)15-s + (0.918 − 0.394i)16-s + (−1.28 + 0.743i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.548 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.729203 + 1.34980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.729203 + 1.34980i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.143 - 1.40i)T \) |
| 3 | \( 1 + (-0.584 - 1.63i)T \) |
| good | 5 | \( 1 + (-2.25 - 0.819i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.05 - 0.715i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (1.59 + 4.39i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.03 + 2.42i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (5.30 - 3.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.49 + 2.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.264 + 1.49i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.88 + 1.58i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-8.06 + 1.42i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (0.515 - 0.297i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.89 - 5.82i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (4.63 - 1.68i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.750 - 4.25i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 6.46T + 53T^{2} \) |
| 59 | \( 1 + (0.703 - 1.93i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.44 - 1.31i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.64 + 4.73i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.78 + 3.08i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.85 - 10.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.66 + 7.94i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (5.79 - 6.90i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (7.85 + 4.53i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.05 - 1.47i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.19308674463717603444122382888, −11.39556049718762108792127224542, −10.57579754753456113111630468673, −9.562293705338195379740806843177, −8.473058632586204384370843705659, −7.999311660793320725485202468463, −6.22837223416078443054860719184, −5.34958028748181042047971124888, −4.48097914404029399315467564638, −2.69732029093376626572360048877,
1.65210698665916350388922091792, 2.28959363763099419488481372935, 4.50895917583805888494507197032, 5.36388776530492998804907001644, 7.06132271185628480780237406309, 8.148767481392373627724975944294, 9.170408061419079353991046175651, 10.01044517978667571365981389449, 11.28182234926127365728924565298, 12.01336138601257668473177664438
