| L(s) = 1 | + 1.41i·2-s − 2.00·4-s + 4.53·5-s − 2.50·7-s − 2.82i·8-s + 6.40i·10-s − 10.4i·11-s + 10.6·13-s − 3.54i·14-s + 4.00·16-s − 23.2·17-s − 10.6i·19-s − 9.06·20-s + 14.7·22-s − 2.83·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.500·4-s + 0.906·5-s − 0.358·7-s − 0.353i·8-s + 0.640i·10-s − 0.950i·11-s + 0.822·13-s − 0.253i·14-s + 0.250·16-s − 1.36·17-s − 0.558i·19-s − 0.453·20-s + 0.671·22-s − 0.123·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.758 + 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06480709797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06480709797\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 \) |
| 113 | \( 1 + (-27.4 + 109. i)T \) |
| good | 5 | \( 1 - 4.53T + 25T^{2} \) |
| 7 | \( 1 + 2.50T + 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 10.6T + 169T^{2} \) |
| 17 | \( 1 + 23.2T + 289T^{2} \) |
| 19 | \( 1 + 10.6iT - 361T^{2} \) |
| 23 | \( 1 + 2.83T + 529T^{2} \) |
| 29 | \( 1 - 8.27T + 841T^{2} \) |
| 31 | \( 1 + 36.1T + 961T^{2} \) |
| 37 | \( 1 + 11.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 74.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.48iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 40.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.68iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 44.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + 57.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 23.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 92.5T + 5.04e3T^{2} \) |
| 73 | \( 1 - 109. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 74.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 86.6iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 120.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 5.77T + 9.40e3T^{2} \) |
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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
−8.671435666295170849420060527341, −7.969215121963338316895100703720, −6.80912521918668670254951878143, −6.30544113479162322730175812178, −5.70118297977790279642970266433, −4.76984227582943887823290484818, −3.75206146287162587394344747621, −2.72041496202855405554159565227, −1.44580796211585214184169203422, −0.01515202631325660213524221666,
1.63219965891257814390647012735, 2.17716607057128792737455262591, 3.39260264939846061169767609042, 4.26505111630073561591287151321, 5.18727343269312221314312299203, 6.09052219636198889965587243278, 6.76034672947115310502812315515, 7.80110794960367534579862760795, 8.811256279020350647970889985520, 9.344824007723141783307109360582
