L(s) = 1 | + 2.27i·2-s − 1.18·4-s − 4.94i·5-s + 2.64·7-s + 6.40i·8-s + 11.2·10-s + 20.1i·11-s + 25.8·13-s + 6.02i·14-s − 19.3·16-s + 6.60i·17-s + 5.53·19-s + 5.86i·20-s − 45.9·22-s − 14.5i·23-s + ⋯ |
L(s) = 1 | + 1.13i·2-s − 0.296·4-s − 0.989i·5-s + 0.377·7-s + 0.800i·8-s + 1.12·10-s + 1.83i·11-s + 1.98·13-s + 0.430i·14-s − 1.20·16-s + 0.388i·17-s + 0.291·19-s + 0.293i·20-s − 2.08·22-s − 0.631i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.23170 + 1.23170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23170 + 1.23170i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 2 | \( 1 - 2.27iT - 4T^{2} \) |
| 5 | \( 1 + 4.94iT - 25T^{2} \) |
| 11 | \( 1 - 20.1iT - 121T^{2} \) |
| 13 | \( 1 - 25.8T + 169T^{2} \) |
| 17 | \( 1 - 6.60iT - 289T^{2} \) |
| 19 | \( 1 - 5.53T + 361T^{2} \) |
| 23 | \( 1 + 14.5iT - 529T^{2} \) |
| 29 | \( 1 + 17.4iT - 841T^{2} \) |
| 31 | \( 1 + 45.3T + 961T^{2} \) |
| 37 | \( 1 - 40.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 30.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 53.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + 27.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 6.45iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 42.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 83.7T + 4.48e3T^{2} \) |
| 71 | \( 1 + 113. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 25.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 28.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.92iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 94.4T + 9.40e3T^{2} \) |
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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
−12.75343733091780277204681041282, −11.70846375880581881133939515913, −10.59939764489005866994978594001, −9.163922185929164441102216794713, −8.400708123112128691418183421594, −7.46130425519915196470169705058, −6.34391543027134882958315670546, −5.25385544539426153523793331483, −4.22695234719187201905166543278, −1.73358107548731186783109328337,
1.23281831665402012666621442971, 3.03752195309893325645102117568, 3.68475919885661763436679225156, 5.78501057921479684029779847488, 6.77383357581420230901213437528, 8.221709930733738848341683531296, 9.290004305173692406206289377951, 10.66883248558239580754654905695, 11.12365586321122304059770941170, 11.55524674874536728844744180116