L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s + 0.999·6-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (−1.70 + 2.95i)11-s + (−0.499 − 0.866i)12-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (0.707 − 1.22i)17-s + (−0.499 + 0.866i)18-s + (−1.41 − 2.44i)19-s + 0.999·20-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s + 0.408·6-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (−0.158 + 0.273i)10-s + (−0.514 + 0.891i)11-s + (−0.144 − 0.249i)12-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.171 − 0.297i)17-s + (−0.117 + 0.204i)18-s + (−0.324 − 0.561i)19-s + 0.223·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7698953506\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7698953506\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1.70 - 2.95i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.414 - 0.717i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.242T + 29T^{2} \) |
| 31 | \( 1 + (-4.53 + 7.85i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.707 + 1.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 7.41T + 43T^{2} \) |
| 47 | \( 1 + (2.53 + 4.39i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.65 - 11.5i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-7.24 + 12.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.171 - 0.297i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.94 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 + (-1.82 + 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.65 + 9.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 + (5.24 + 9.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−9.552909842787325594749067748140, −8.627921649640137616815804712483, −7.82872520794894513050446717953, −7.01496126336090822623716916298, −5.83863340725434712325054509560, −4.81510848255930719762375182890, −4.27284539593529618821790422922, −3.10427964429370067315110536756, −1.99958523418173970202159386959, −0.41727357623473776281040838839,
1.11112538662217581010700687359, 2.59711669259207851235398988256, 3.75591419435204632760761275178, 5.00993183306606390268070222634, 5.82036260820054687766156706558, 6.55644702096843847254803467350, 7.25989299742626737596903775879, 8.246410669436256660356223918726, 8.495021314128241255974282419760, 9.754810330913411151507361382501