| L(s) = 1 | − 1.41·2-s + 2.00·4-s − 4.46i·5-s − 2.82·8-s + 6.30i·10-s − 2.82·11-s + 18.6i·13-s + 4.00·16-s − 12.0i·17-s − 13.8i·19-s − 8.92i·20-s + 4.00·22-s + 36.4·23-s + 5.10·25-s − 26.3i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.892i·5-s − 0.353·8-s + 0.630i·10-s − 0.257·11-s + 1.43i·13-s + 0.250·16-s − 0.708i·17-s − 0.730i·19-s − 0.446i·20-s + 0.181·22-s + 1.58·23-s + 0.204·25-s − 1.01i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.144963797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.144963797\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.46iT - 25T^{2} \) |
| 11 | \( 1 + 2.82T + 121T^{2} \) |
| 13 | \( 1 - 18.6iT - 169T^{2} \) |
| 17 | \( 1 + 12.0iT - 289T^{2} \) |
| 19 | \( 1 + 13.8iT - 361T^{2} \) |
| 23 | \( 1 - 36.4T + 529T^{2} \) |
| 29 | \( 1 - 12.4T + 841T^{2} \) |
| 31 | \( 1 - 15.1iT - 961T^{2} \) |
| 37 | \( 1 - 45.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 71.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 84.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 104.T + 2.80e3T^{2} \) |
| 59 | \( 1 - 24.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 13.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 92.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + 85.4iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 3.59T + 6.24e3T^{2} \) |
| 83 | \( 1 + 80.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 23.0iT - 9.40e3T^{2} \) |
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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
−9.419113865767098178791596291219, −9.038241192966179838062991194424, −8.307919362140517767212078204830, −7.15689661532790408107116481947, −6.61286725052009565089526291812, −5.20096923423759889207768243566, −4.56660200105487468940118985797, −3.07410455198320801238847409696, −1.78110355783113400873418940804, −0.55422460918542697101666641065,
1.12132898941795147926253189414, 2.68986230021486152592500137760, 3.36049015038990907232615518406, 4.92762998568177963923305362654, 6.05918785485257066675231171958, 6.73649062666131920809098873022, 7.85870049788633334589932504677, 8.183640243275569293756068863360, 9.464492070063359993620907469533, 10.12744502700263095749504034850
