| L(s) = 1 | + (−2.76 − 1.16i)3-s + (−0.148 + 0.110i)5-s + (5.09 − 3.35i)7-s + (6.28 + 6.44i)9-s + (−1.69 + 14.4i)11-s + (−2.59 − 8.67i)13-s + (0.538 − 0.132i)15-s + (6.34 + 1.11i)17-s + (−4.95 − 28.1i)19-s + (−18.0 + 3.32i)21-s + (24.7 − 37.6i)23-s + (−7.16 + 23.9i)25-s + (−9.85 − 25.1i)27-s + (35.2 − 33.2i)29-s + (0.345 − 5.92i)31-s + ⋯ |
| L(s) = 1 | + (−0.921 − 0.388i)3-s + (−0.0296 + 0.0220i)5-s + (0.728 − 0.479i)7-s + (0.698 + 0.715i)9-s + (−0.153 + 1.31i)11-s + (−0.199 − 0.667i)13-s + (0.0358 − 0.00881i)15-s + (0.373 + 0.0658i)17-s + (−0.260 − 1.47i)19-s + (−0.857 + 0.158i)21-s + (1.07 − 1.63i)23-s + (−0.286 + 0.956i)25-s + (−0.365 − 0.930i)27-s + (1.21 − 1.14i)29-s + (0.0111 − 0.191i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.03359 - 0.641699i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03359 - 0.641699i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.76 + 1.16i)T \) |
| good | 5 | \( 1 + (0.148 - 0.110i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-5.09 + 3.35i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (1.69 - 14.4i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (2.59 + 8.67i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (-6.34 - 1.11i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (4.95 + 28.1i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-24.7 + 37.6i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (-35.2 + 33.2i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-0.345 + 5.92i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (-59.7 + 21.7i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (9.46 - 39.9i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (33.4 + 77.6i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (8.86 - 0.516i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (-16.9 - 9.78i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (1.86 + 15.9i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (61.9 + 31.0i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (-12.7 + 13.4i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-49.6 + 59.1i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (51.1 - 42.9i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (-95.3 + 22.6i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (-35.6 - 150. i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-64.2 - 76.5i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (-61.2 + 82.2i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−11.15927917126007282596965567963, −10.54691020114562427082879529000, −9.578449798989778104178999268216, −8.106376288732160659215698387682, −7.30234712608467754950300047268, −6.47381691125800032520957927520, −5.03385969946967081366996652632, −4.50940192255732099771840036773, −2.37769805468740035718904783924, −0.75747909613807939706186998865,
1.30151042643867354792217285856, 3.33795947208944638556712035306, 4.68439438198756637835431442342, 5.60427272005315546068736128849, 6.43307681211391228734812732545, 7.81242212062084458019988792604, 8.770603140814311818804291643340, 9.845392356477863442444355945567, 10.78138359138316953815524996323, 11.59298570127098779006178446771
