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
−10.12744502700263095749504034850, −9.464492070063359993620907469533, −8.183640243275569293756068863360, −7.85870049788633334589932504677, −6.73649062666131920809098873022, −6.05918785485257066675231171958, −4.92762998568177963923305362654, −3.36049015038990907232615518406, −2.68986230021486152592500137760, −1.12132898941795147926253189414,
0.55422460918542697101666641065, 1.78110355783113400873418940804, 3.07410455198320801238847409696, 4.56660200105487468940118985797, 5.20096923423759889207768243566, 6.61286725052009565089526291812, 7.15689661532790408107116481947, 8.307919362140517767212078204830, 9.038241192966179838062991194424, 9.419113865767098178791596291219