| L(s) = 1 | + (−4.25 − 2.63i)5-s + 2.72i·7-s − 3.20i·11-s − 7.99i·13-s − 1.82·17-s − 15.4·19-s − 32.3·23-s + (11.1 + 22.3i)25-s + 29.5i·29-s + 53.0·31-s + (7.16 − 11.5i)35-s + 30.0i·37-s − 76.8i·41-s − 8.98i·43-s + 11.8·47-s + ⋯ |
| L(s) = 1 | + (−0.850 − 0.526i)5-s + 0.388i·7-s − 0.291i·11-s − 0.614i·13-s − 0.107·17-s − 0.812·19-s − 1.40·23-s + (0.445 + 0.895i)25-s + 1.01i·29-s + 1.70·31-s + (0.204 − 0.330i)35-s + 0.812i·37-s − 1.87i·41-s − 0.208i·43-s + 0.252·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.037294401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.037294401\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.25 + 2.63i)T \) |
| good | 7 | \( 1 - 2.72iT - 49T^{2} \) |
| 11 | \( 1 + 3.20iT - 121T^{2} \) |
| 13 | \( 1 + 7.99iT - 169T^{2} \) |
| 17 | \( 1 + 1.82T + 289T^{2} \) |
| 19 | \( 1 + 15.4T + 361T^{2} \) |
| 23 | \( 1 + 32.3T + 529T^{2} \) |
| 29 | \( 1 - 29.5iT - 841T^{2} \) |
| 31 | \( 1 - 53.0T + 961T^{2} \) |
| 37 | \( 1 - 30.0iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 76.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.98iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 11.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 28.9T + 2.80e3T^{2} \) |
| 59 | \( 1 + 19.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 61.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 5.18iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 96.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 127. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 48.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 96.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 21.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 166. iT - 9.40e3T^{2} \) |
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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
−9.140879858128684352120962669478, −8.430622608355075500481293966495, −7.986205576832009140885859081876, −6.98446887208006210212020386210, −6.04742436892100756599115664017, −5.18817068393339956718787750420, −4.29296560548113534463855024222, −3.45669089922723748889333127569, −2.30534018045693642938460774711, −0.864998749802167636157090369126,
0.36518278996770575546338407309, 1.97230368144562252948066039632, 3.06142931276521010486438314872, 4.22134048369247911251363906587, 4.51634521398836471952690177354, 6.13860468053606291745381217195, 6.57803097041980307338172466213, 7.67978062836356286787900808392, 8.028074629787264589004891008016, 9.059523300036182097253453726203
