| L(s) = 1 | + (−1.37 − 0.568i)3-s + (−2.28 + 0.948i)5-s + (6.37 + 6.37i)7-s + (−4.80 − 4.80i)9-s + (−1.79 + 0.744i)11-s + (−16.7 − 6.91i)13-s + 3.68·15-s + 6.19i·17-s + (−8.50 + 20.5i)19-s + (−5.12 − 12.3i)21-s + (−23.6 + 23.6i)23-s + (−13.3 + 13.3i)25-s + (8.98 + 21.6i)27-s + (−14.5 + 35.1i)29-s − 14.1i·31-s + ⋯ |
| L(s) = 1 | + (−0.457 − 0.189i)3-s + (−0.457 + 0.189i)5-s + (0.911 + 0.911i)7-s + (−0.533 − 0.533i)9-s + (−0.163 + 0.0676i)11-s + (−1.28 − 0.532i)13-s + 0.245·15-s + 0.364i·17-s + (−0.447 + 1.08i)19-s + (−0.244 − 0.589i)21-s + (−1.02 + 1.02i)23-s + (−0.533 + 0.533i)25-s + (0.332 + 0.802i)27-s + (−0.502 + 1.21i)29-s − 0.456i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.150574 + 0.433680i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.150574 + 0.433680i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (1.37 + 0.568i)T + (6.36 + 6.36i)T^{2} \) |
| 5 | \( 1 + (2.28 - 0.948i)T + (17.6 - 17.6i)T^{2} \) |
| 7 | \( 1 + (-6.37 - 6.37i)T + 49iT^{2} \) |
| 11 | \( 1 + (1.79 - 0.744i)T + (85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (16.7 + 6.91i)T + (119. + 119. i)T^{2} \) |
| 17 | \( 1 - 6.19iT - 289T^{2} \) |
| 19 | \( 1 + (8.50 - 20.5i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (23.6 - 23.6i)T - 529iT^{2} \) |
| 29 | \( 1 + (14.5 - 35.1i)T + (-594. - 594. i)T^{2} \) |
| 31 | \( 1 + 14.1iT - 961T^{2} \) |
| 37 | \( 1 + (-30.0 + 12.4i)T + (968. - 968. i)T^{2} \) |
| 41 | \( 1 + (56.9 + 56.9i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-54.5 + 22.5i)T + (1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 34.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + (3.92 + 9.48i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-9.41 - 22.7i)T + (-2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (3.00 - 7.25i)T + (-2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (55.9 + 23.1i)T + (3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (6.27 + 6.27i)T + 5.04e3iT^{2} \) |
| 73 | \( 1 + (-66.4 - 66.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 75.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + (1.23 - 2.97i)T + (-4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-36.7 + 36.7i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 - 90.0T + 9.40e3T^{2} \) |
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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
−12.04298095513498222531877695207, −11.47151259168414254998865292801, −10.40441230930742119191187255526, −9.233879057344686005410878714314, −8.151920432243355970950183962593, −7.37361459862299920138599106765, −5.88388390933781533518382882867, −5.23284163467066755168280558605, −3.64110011305494186388894911465, −2.02540205831753458758894901213,
0.23684755750913011653928556828, 2.37485660037964185500663483904, 4.41255143069232551395649674674, 4.83512220700642068958901653630, 6.37757790351874975560519995467, 7.64034470823235704507468792319, 8.232108971703519103663899507168, 9.663431100946401411664440205236, 10.62947337094664349639784341930, 11.44158431885628564813949508314
