L(s) = 1 | + 0.547i·3-s + (−3.30 − 3.75i)5-s − 10.0·7-s + 8.69·9-s − 17.2·11-s + 4.41·13-s + (2.05 − 1.80i)15-s + 27.0i·17-s + 4.82·19-s − 5.50i·21-s − 15.2·23-s + (−3.19 + 24.7i)25-s + 9.69i·27-s + 2.38i·29-s − 38.0i·31-s + ⋯ |
L(s) = 1 | + 0.182i·3-s + (−0.660 − 0.750i)5-s − 1.43·7-s + 0.966·9-s − 1.56·11-s + 0.339·13-s + (0.137 − 0.120i)15-s + 1.58i·17-s + 0.254·19-s − 0.262i·21-s − 0.663·23-s + (−0.127 + 0.991i)25-s + 0.359i·27-s + 0.0821i·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0639i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.035377500\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.035377500\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.30 + 3.75i)T \) |
good | 3 | \( 1 - 0.547iT - 9T^{2} \) |
| 7 | \( 1 + 10.0T + 49T^{2} \) |
| 11 | \( 1 + 17.2T + 121T^{2} \) |
| 13 | \( 1 - 4.41T + 169T^{2} \) |
| 17 | \( 1 - 27.0iT - 289T^{2} \) |
| 19 | \( 1 - 4.82T + 361T^{2} \) |
| 23 | \( 1 + 15.2T + 529T^{2} \) |
| 29 | \( 1 - 2.38iT - 841T^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + 16.5T + 1.36e3T^{2} \) |
| 41 | \( 1 - 13.3T + 1.68e3T^{2} \) |
| 43 | \( 1 + 59.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 62.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 71.5T + 2.80e3T^{2} \) |
| 59 | \( 1 - 68.8T + 3.48e3T^{2} \) |
| 61 | \( 1 + 40.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 51.0iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 40.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 35.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 126. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 75.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 106.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 85.4iT - 9.40e3T^{2} \) |
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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
−9.549645271006248326875227845833, −8.636988703745940828139235594281, −7.86747095240359003811698634683, −7.13172443678780779541161320510, −6.05496846993869401488320485751, −5.28902403434971863835261007505, −4.06797122011316538140973190243, −3.60857991710235021639806646737, −2.19047629754779823378937246228, −0.58453115693454411583018332134,
0.56538060070946042513462071153, 2.53030477948538587832540220296, 3.16827804783826439664183654024, 4.18569609139991169348310433943, 5.31662298870362110898942676827, 6.36679245640613792433815707973, 7.19649639345073609216029318719, 7.51678228051480559596517956742, 8.642196825793219471306718089430, 9.757284262533663960258105501138