| L(s) = 1 | − 4.41i·5-s + 3.59i·7-s + 18.7·11-s − 19.9i·13-s − 8.00·17-s − 16.1·19-s + 18.3i·23-s + 5.52·25-s + 17.9i·29-s − 29.7i·31-s + 15.8·35-s − 26.7i·37-s − 40.2·41-s + 71.8·43-s − 23.3i·47-s + ⋯ |
| L(s) = 1 | − 0.882i·5-s + 0.514i·7-s + 1.70·11-s − 1.53i·13-s − 0.470·17-s − 0.849·19-s + 0.799i·23-s + 0.221·25-s + 0.618i·29-s − 0.959i·31-s + 0.453·35-s − 0.722i·37-s − 0.981·41-s + 1.67·43-s − 0.497i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.802458988\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.802458988\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4.41iT - 25T^{2} \) |
| 7 | \( 1 - 3.59iT - 49T^{2} \) |
| 11 | \( 1 - 18.7T + 121T^{2} \) |
| 13 | \( 1 + 19.9iT - 169T^{2} \) |
| 17 | \( 1 + 8.00T + 289T^{2} \) |
| 19 | \( 1 + 16.1T + 361T^{2} \) |
| 23 | \( 1 - 18.3iT - 529T^{2} \) |
| 29 | \( 1 - 17.9iT - 841T^{2} \) |
| 31 | \( 1 + 29.7iT - 961T^{2} \) |
| 37 | \( 1 + 26.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 40.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 71.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + 23.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 90.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 10.9T + 3.48e3T^{2} \) |
| 61 | \( 1 + 90.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 74.0T + 4.48e3T^{2} \) |
| 71 | \( 1 - 13.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 56.0T + 5.32e3T^{2} \) |
| 79 | \( 1 + 118. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 8.08T + 6.88e3T^{2} \) |
| 89 | \( 1 - 83.9T + 7.92e3T^{2} \) |
| 97 | \( 1 + 79.0T + 9.40e3T^{2} \) |
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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
−9.565235736088068958827182798552, −8.908605072189585544869161516801, −8.329840000822982499520782755676, −7.22354928858917173994835017914, −6.16822373732651865956643732439, −5.41264696235356207119734972653, −4.38201279303271366634610430721, −3.39599102791129238506664975522, −1.91439659926829837539694609842, −0.64604715659655291337107970890,
1.35299015711061056848323165980, 2.61975544432949481400788971866, 3.99753342737197226952362185897, 4.43153892432626206736395381359, 6.22057299276492352345765975478, 6.67428548513591234002042794845, 7.30569648861811987071089284550, 8.732808275889607206447192057459, 9.171261408436406363687055382439, 10.29420718985667590512428542598
