L(s) = 1 | + (2.12 − 2.12i)2-s − 0.999i·4-s + (9 + 9i)7-s + (14.8 + 14.8i)8-s + 38.1i·11-s + (−63 + 63i)13-s + 38.1·14-s + 71·16-s + (29.6 − 29.6i)17-s − 70i·19-s + (81 + 81i)22-s + (72.1 + 72.1i)23-s + 267. i·26-s + (8.99 − 8.99i)28-s + 229.·29-s + ⋯ |
L(s) = 1 | + (0.749 − 0.749i)2-s − 0.124i·4-s + (0.485 + 0.485i)7-s + (0.656 + 0.656i)8-s + 1.04i·11-s + (−1.34 + 1.34i)13-s + 0.728·14-s + 1.10·16-s + (0.423 − 0.423i)17-s − 0.845i·19-s + (0.784 + 0.784i)22-s + (0.653 + 0.653i)23-s + 2.01i·26-s + (0.0607 − 0.0607i)28-s + 1.46·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.60059 + 0.529526i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.60059 + 0.529526i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-2.12 + 2.12i)T - 8iT^{2} \) |
| 7 | \( 1 + (-9 - 9i)T + 343iT^{2} \) |
| 11 | \( 1 - 38.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (63 - 63i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (-29.6 + 29.6i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 70iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-72.1 - 72.1i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196T + 2.97e4T^{2} \) |
| 37 | \( 1 + (207 + 207i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 267. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (-144 + 144i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (356. - 356. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (224. + 224. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 + (378 + 378i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 840. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (378 - 378i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 488iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-772. - 772. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 267.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-252 - 252i)T + 9.12e5iT^{2} \) |
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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
−11.97892381624203378926269822541, −11.31230734723030902393456570442, −10.04496107318974126999244218825, −9.104349985249497662062405843251, −7.77534750555935756620399908332, −6.82890121268232047062070593990, −4.95743222767456540934319595443, −4.58347702324232167461214952940, −2.87739842452781200975224862272, −1.85938931847137398739171201743,
0.919806287717927038179160568658, 3.13360385758805915106042077520, 4.58288627526350190147756870882, 5.44249788431659229995224859387, 6.46362954514640221937282889832, 7.61329339867146147465158685342, 8.384196639544068837722160842646, 10.15898584374375638078024652546, 10.50119503954218261036358904386, 11.99490302358475737881751583426