L(s) = 1 | + (4.58 + 2.00i)5-s + (9.28 − 5.36i)7-s + (10.5 − 6.09i)11-s + (20.2 + 11.6i)13-s + 1.67·17-s + 0.234·19-s + (−9.93 + 17.2i)23-s + (16.9 + 18.3i)25-s + (29.5 − 17.0i)29-s + (9.53 − 16.5i)31-s + (53.2 − 5.97i)35-s + 32.7i·37-s + (−4.57 − 2.63i)41-s + (−45.8 + 26.4i)43-s + (−26.8 − 46.5i)47-s + ⋯ |
L(s) = 1 | + (0.916 + 0.400i)5-s + (1.32 − 0.766i)7-s + (0.959 − 0.553i)11-s + (1.55 + 0.899i)13-s + 0.0987·17-s + 0.0123·19-s + (−0.431 + 0.747i)23-s + (0.679 + 0.733i)25-s + (1.01 − 0.587i)29-s + (0.307 − 0.532i)31-s + (1.52 − 0.170i)35-s + 0.884i·37-s + (−0.111 − 0.0643i)41-s + (−1.06 + 0.616i)43-s + (−0.571 − 0.989i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0244i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.437455862\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.437455862\) |
\(L(2)\) |
|
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 | \( 1 + (-4.58 - 2.00i)T \) |
good | 7 | \( 1 + (-9.28 + 5.36i)T + (24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-10.5 + 6.09i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-20.2 - 11.6i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 1.67T + 289T^{2} \) |
| 19 | \( 1 - 0.234T + 361T^{2} \) |
| 23 | \( 1 + (9.93 - 17.2i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-29.5 + 17.0i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-9.53 + 16.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 32.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (4.57 + 2.63i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (45.8 - 26.4i)T + (924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (26.8 + 46.5i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 84.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (76.4 + 44.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-32.7 - 56.8i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (86.1 + 49.7i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 36.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 79.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (17.6 + 30.6i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-13.9 - 24.1i)T + (-3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 152. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-84.0 + 48.5i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.181022413250645277517920267944, −8.428571226918719299107759021707, −7.71817817538014497077617481389, −6.46490625575387505724555006126, −6.28916858988688032730667318194, −5.04434848865852942278765711748, −4.17282594699870395873638316459, −3.28581429374927791363833491230, −1.71881642924458199926522758209, −1.25094450445169527977594661963,
1.20341187801446996272838074614, 1.80966210262704832970127002087, 3.06673242682226801536880082920, 4.39461535354934330157625871665, 5.09474774640817420355117702763, 5.95774362688555622592627337250, 6.54138355776885093698344952372, 7.85182873695804205828086086386, 8.619653908594434654396592957077, 8.922657084359277563245866821039