L(s) = 1 | + (−0.686 − 2.92i)3-s + (−6.55 + 3.78i)5-s + (4.55 − 7.89i)7-s + (−8.05 + 4.00i)9-s + (−0.383 − 0.221i)11-s + (−5.55 − 9.62i)13-s + (15.5 + 16.5i)15-s + 8.01i·17-s − 8.11·19-s + (−26.1 − 7.89i)21-s + (−20.4 + 11.8i)23-s + (16.1 − 28.0i)25-s + (17.2 + 20.7i)27-s + (45.9 + 26.5i)29-s + (14.6 + 25.4i)31-s + ⋯ |
L(s) = 1 | + (−0.228 − 0.973i)3-s + (−1.31 + 0.757i)5-s + (0.651 − 1.12i)7-s + (−0.895 + 0.445i)9-s + (−0.0348 − 0.0201i)11-s + (−0.427 − 0.740i)13-s + (1.03 + 1.10i)15-s + 0.471i·17-s − 0.427·19-s + (−1.24 − 0.375i)21-s + (−0.888 + 0.513i)23-s + (0.647 − 1.12i)25-s + (0.638 + 0.769i)27-s + (1.58 + 0.913i)29-s + (0.473 + 0.819i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5050657867\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5050657867\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.686 + 2.92i)T \) |
good | 5 | \( 1 + (6.55 - 3.78i)T + (12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + (-4.55 + 7.89i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (0.383 + 0.221i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (5.55 + 9.62i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 - 8.01iT - 289T^{2} \) |
| 19 | \( 1 + 8.11T + 361T^{2} \) |
| 23 | \( 1 + (20.4 - 11.8i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-45.9 - 26.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-14.6 - 25.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 18.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + (38.9 - 22.4i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (11.5 - 19.9i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-7.32 - 4.22i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 60.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-65.9 + 38.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-2.67 + 4.63i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-54.8 - 95.0i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 16.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 4.35T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-0.792 + 1.37i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (7.32 + 4.22i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + 64.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (57.6 - 99.7i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−10.72657193068574839327521959883, −10.30966942164301887687609879736, −8.357905808808089418759312331421, −7.984552884283637606343944062469, −7.16339103816803515449075894351, −6.55846744460646930940636159180, −5.10077246551337794307658728208, −3.96608926035388466838536266855, −2.87577714873654732070445854484, −1.18675262132833167762296083529,
0.22276757648975246295776749337, 2.40799943848669390954191755520, 3.92107054738834698833586777619, 4.63724361381023645291200817643, 5.36010100822824520991443427932, 6.62277375253424757862300992004, 8.148785018295933471387084680797, 8.459354490252331849527759513841, 9.362765809278932739723777683496, 10.32237675253564246757470624329