| L(s) = 1 | + (−2.12 + 2.12i)3-s + (10.2 + 10.2i)5-s − 32.8i·7-s − 8.99i·9-s + (18.2 + 18.2i)11-s + (22.5 − 22.5i)13-s − 43.6·15-s + 50.1·17-s + (6.68 − 6.68i)19-s + (69.6 + 69.6i)21-s + 186. i·23-s + 86.9i·25-s + (19.0 + 19.0i)27-s + (118. − 118. i)29-s + 250.·31-s + ⋯ |
| L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.920 + 0.920i)5-s − 1.77i·7-s − 0.333i·9-s + (0.499 + 0.499i)11-s + (0.481 − 0.481i)13-s − 0.751·15-s + 0.715·17-s + (0.0807 − 0.0807i)19-s + (0.723 + 0.723i)21-s + 1.69i·23-s + 0.695i·25-s + (0.136 + 0.136i)27-s + (0.757 − 0.757i)29-s + 1.44·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.972 - 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.86760 + 0.222282i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86760 + 0.222282i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.12 - 2.12i)T \) |
| good | 5 | \( 1 + (-10.2 - 10.2i)T + 125iT^{2} \) |
| 7 | \( 1 + 32.8iT - 343T^{2} \) |
| 11 | \( 1 + (-18.2 - 18.2i)T + 1.33e3iT^{2} \) |
| 13 | \( 1 + (-22.5 + 22.5i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 - 50.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + (-6.68 + 6.68i)T - 6.85e3iT^{2} \) |
| 23 | \( 1 - 186. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-118. + 118. i)T - 2.43e4iT^{2} \) |
| 31 | \( 1 - 250.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-198. - 198. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + 186. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (10.9 + 10.9i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + 23.1T + 1.03e5T^{2} \) |
| 53 | \( 1 + (-134. - 134. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (220. + 220. i)T + 2.05e5iT^{2} \) |
| 61 | \( 1 + (453. - 453. i)T - 2.26e5iT^{2} \) |
| 67 | \( 1 + (-184. + 184. i)T - 3.00e5iT^{2} \) |
| 71 | \( 1 - 18.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 + 828. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-173. + 173. i)T - 5.71e5iT^{2} \) |
| 89 | \( 1 - 335. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 687.T + 9.12e5T^{2} \) |
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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
−11.91457306914759691722354951664, −10.87752542808031958816548644988, −10.14386689458572180662457316856, −9.674068626271713981187565542617, −7.81127465555261005198073397859, −6.83378606923692136195495399250, −5.92845821429848905568155069390, −4.42372222238006911923707267494, −3.23644530275392427162874076793, −1.17115417920123051463198753076,
1.22985904211299019388564127945, 2.59149774875863120990447689040, 4.79239307763165928644153129395, 5.82059503163988142842086285778, 6.40050629393269791190590940252, 8.367107948463225690239847888736, 8.892068785850589684225691500385, 9.894932166491974606330803058037, 11.32441601805881099253759135407, 12.25318701960811419585080361535
