L(s) = 1 | + (2.45 − 0.962i)2-s + (3.61 − 3.35i)4-s + (−1.47 + 1.00i)5-s + (2.09 + 1.60i)7-s + (3.35 − 6.97i)8-s + (−2.65 + 3.89i)10-s + (0.538 − 3.57i)11-s + (−2.76 − 2.20i)13-s + (6.69 + 1.92i)14-s + (0.785 − 10.4i)16-s + (−2.32 + 0.718i)17-s + (6.52 + 3.76i)19-s + (−1.96 + 8.60i)20-s + (−2.11 − 9.28i)22-s + (−1.91 + 6.21i)23-s + ⋯ |
L(s) = 1 | + (1.73 − 0.680i)2-s + (1.80 − 1.67i)4-s + (−0.660 + 0.450i)5-s + (0.793 + 0.608i)7-s + (1.18 − 2.46i)8-s + (−0.839 + 1.23i)10-s + (0.162 − 1.07i)11-s + (−0.767 − 0.612i)13-s + (1.79 + 0.514i)14-s + (0.196 − 2.61i)16-s + (−0.564 + 0.174i)17-s + (1.49 + 0.864i)19-s + (−0.439 + 1.92i)20-s + (−0.451 − 1.97i)22-s + (−0.399 + 1.29i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.07486 - 1.62067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.07486 - 1.62067i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-2.09 - 1.60i)T \) |
good | 2 | \( 1 + (-2.45 + 0.962i)T + (1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (1.47 - 1.00i)T + (1.82 - 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.538 + 3.57i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (2.76 + 2.20i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (2.32 - 0.718i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-6.52 - 3.76i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.91 - 6.21i)T + (-19.0 - 12.9i)T^{2} \) |
| 29 | \( 1 + (4.35 + 0.994i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (5.85 - 3.38i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.18 + 3.88i)T + (2.76 + 36.8i)T^{2} \) |
| 41 | \( 1 + (3.53 + 1.70i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-3.05 + 1.47i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (1.81 + 4.62i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.20 - 3.45i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (9.48 + 6.46i)T + (21.5 + 54.9i)T^{2} \) |
| 61 | \( 1 + (-4.54 + 4.89i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-6.90 - 11.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.21 - 0.276i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.19 - 3.21i)T + (53.5 + 49.6i)T^{2} \) |
| 79 | \( 1 + (-5.24 + 9.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.72 - 10.9i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-7.03 + 1.05i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 - 1.36iT - 97T^{2} \) |
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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
−11.31343080389157514205563953917, −10.70120158738112689260908174702, −9.454177455482616623868972316806, −7.958031106048426305047489688557, −7.07864908161679412750314606382, −5.55284325426979328617464869997, −5.39197336578676626091852420691, −3.82334367785787589127387299198, −3.21312512019983604534119237306, −1.81603406776416361881767420367,
2.23807741730123195996622451639, 3.81772592128278626560516376289, 4.65925218708779713571247910718, 5.05732537848423196077725656123, 6.62440036513654985790908270348, 7.34168643394737426244997912620, 7.945127971219333717066920865853, 9.342869384547162987674927796944, 10.83116526128288318766026633863, 11.79628580492236885960575730877