| L(s) = 1 | + (−0.429 − 1.60i)2-s + (−4.91 + 1.67i)3-s + (4.54 − 2.62i)4-s + (−7.92 − 7.88i)5-s + (4.79 + 7.15i)6-s + (−30.4 + 8.15i)7-s + (−15.5 − 15.5i)8-s + (21.3 − 16.5i)9-s + (−9.22 + 16.0i)10-s + (−31.0 − 17.9i)11-s + (−17.9 + 20.5i)12-s + (53.6 + 14.3i)13-s + (26.1 + 45.2i)14-s + (52.2 + 25.4i)15-s + (2.79 − 4.84i)16-s + (29.9 − 29.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.151 − 0.566i)2-s + (−0.946 + 0.323i)3-s + (0.568 − 0.328i)4-s + (−0.709 − 0.705i)5-s + (0.326 + 0.486i)6-s + (−1.64 + 0.440i)7-s + (−0.686 − 0.686i)8-s + (0.791 − 0.611i)9-s + (−0.291 + 0.508i)10-s + (−0.850 − 0.490i)11-s + (−0.431 + 0.494i)12-s + (1.14 + 0.306i)13-s + (0.498 + 0.863i)14-s + (0.898 + 0.437i)15-s + (0.0436 − 0.0756i)16-s + (0.427 − 0.427i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.252i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0605723 - 0.472777i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0605723 - 0.472777i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.91 - 1.67i)T \) |
| 5 | \( 1 + (7.92 + 7.88i)T \) |
| good | 2 | \( 1 + (0.429 + 1.60i)T + (-6.92 + 4i)T^{2} \) |
| 7 | \( 1 + (30.4 - 8.15i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (31.0 + 17.9i)T + (665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-53.6 - 14.3i)T + (1.90e3 + 1.09e3i)T^{2} \) |
| 17 | \( 1 + (-29.9 + 29.9i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 17.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-27.3 + 101. i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (26.5 - 46.0i)T + (-1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (19.1 + 33.0i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (75.1 + 75.1i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (126. - 73.2i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (140. + 526. i)T + (-6.88e4 + 3.97e4i)T^{2} \) |
| 47 | \( 1 + (-43.2 - 161. i)T + (-8.99e4 + 5.19e4i)T^{2} \) |
| 53 | \( 1 + (170. + 170. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-28.0 - 48.5i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-187. + 324. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (29.6 - 110. i)T + (-2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 - 921. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-373. + 373. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (492. + 284. i)T + (2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-439. + 117. i)T + (4.95e5 - 2.85e5i)T^{2} \) |
| 89 | \( 1 + 148.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (312. - 83.7i)T + (7.90e5 - 4.56e5i)T^{2} \) |
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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
−15.49165965209062425123670630711, −13.06439752074237127014853382970, −12.28532088323798302349541910058, −11.25399049112192753364031287124, −10.20199488896984318250261728557, −8.994256806202453814770983284916, −6.77008872752096467897260479560, −5.59678368457373766482898335181, −3.44903592543734264235085039206, −0.42223713707189174002681408533,
3.37216518015876880493302746542, 5.99076634235561013903564161828, 6.89488282059757050025016697120, 7.86919189814650099631721654355, 10.15025708944058599015970270893, 11.13617583721406356117790418281, 12.35482964726354797675733805342, 13.30709705792164870146338376324, 15.33478532439725188661732611078, 15.93883682729458876833120297410
