L(s) = 1 | + (−0.618 − 1.61i)3-s + 3.23i·5-s + 1.23i·7-s + (−2.23 + 2.00i)9-s − 5.23·11-s − 4.47·13-s + (5.23 − 2.00i)15-s + 2.47i·17-s + 0.763i·19-s + (2.00 − 0.763i)21-s + 2.47·23-s − 5.47·25-s + (4.61 + 2.38i)27-s + 4.76i·29-s + 5.23i·31-s + ⋯ |
L(s) = 1 | + (−0.356 − 0.934i)3-s + 1.44i·5-s + 0.467i·7-s + (−0.745 + 0.666i)9-s − 1.57·11-s − 1.24·13-s + (1.35 − 0.516i)15-s + 0.599i·17-s + 0.175i·19-s + (0.436 − 0.166i)21-s + 0.515·23-s − 1.09·25-s + (0.888 + 0.458i)27-s + 0.884i·29-s + 0.940i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.356 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.354541 + 0.514948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.354541 + 0.514948i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.618 + 1.61i)T \) |
good | 5 | \( 1 - 3.23iT - 5T^{2} \) |
| 7 | \( 1 - 1.23iT - 7T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 19 | \( 1 - 0.763iT - 19T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 - 4.76iT - 29T^{2} \) |
| 31 | \( 1 - 5.23iT - 31T^{2} \) |
| 37 | \( 1 - 8.47T + 37T^{2} \) |
| 41 | \( 1 + 6.47iT - 41T^{2} \) |
| 43 | \( 1 + 7.23iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 3.23iT - 53T^{2} \) |
| 59 | \( 1 - 1.23T + 59T^{2} \) |
| 61 | \( 1 - 0.472T + 61T^{2} \) |
| 67 | \( 1 - 9.70iT - 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 0.291iT - 79T^{2} \) |
| 83 | \( 1 + 2.76T + 83T^{2} \) |
| 89 | \( 1 + 4iT - 89T^{2} \) |
| 97 | \( 1 - 0.472T + 97T^{2} \) |
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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
−11.61334535443461541692309507957, −10.66143657439207703535620966463, −10.21553188661644303091822773242, −8.649850720816424690474274811493, −7.52962059160023067121229253521, −7.09850355644598100497572677221, −5.96930643351936550636399437278, −5.08143475494559960397632849984, −3.00485006546409638162381684884, −2.27482523376426099150410881987,
0.40534845052472367622557660495, 2.77065523905238320012856226399, 4.53068040854485427414322890199, 4.86746939517290063123510886470, 5.87214610005393243535764031529, 7.52827577989536865755828693224, 8.337383290270648066727179239920, 9.543420385469248646714315408945, 9.886303012611847357393408128695, 11.05305105836658184844234775186