L(s) = 1 | + 5.46·3-s + (−3.35 + 3.70i)5-s + 5.17·7-s + 20.8·9-s − 3.02i·11-s + 19.8i·13-s + (−18.3 + 20.2i)15-s + 29.5i·17-s − 19.7i·19-s + 28.2·21-s − 6.31·23-s + (−2.52 − 24.8i)25-s + 64.7·27-s − 19.8·29-s − 19.3i·31-s + ⋯ |
L(s) = 1 | + 1.82·3-s + (−0.670 + 0.741i)5-s + 0.738·7-s + 2.31·9-s − 0.275i·11-s + 1.52i·13-s + (−1.22 + 1.35i)15-s + 1.73i·17-s − 1.03i·19-s + 1.34·21-s − 0.274·23-s + (−0.100 − 0.994i)25-s + 2.39·27-s − 0.683·29-s − 0.623i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.353413404\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.353413404\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (3.35 - 3.70i)T \) |
good | 3 | \( 1 - 5.46T + 9T^{2} \) |
| 7 | \( 1 - 5.17T + 49T^{2} \) |
| 11 | \( 1 + 3.02iT - 121T^{2} \) |
| 13 | \( 1 - 19.8iT - 169T^{2} \) |
| 17 | \( 1 - 29.5iT - 289T^{2} \) |
| 19 | \( 1 + 19.7iT - 361T^{2} \) |
| 23 | \( 1 + 6.31T + 529T^{2} \) |
| 29 | \( 1 + 19.8T + 841T^{2} \) |
| 31 | \( 1 + 19.3iT - 961T^{2} \) |
| 37 | \( 1 + 7.92iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 66.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 8.67T + 1.84e3T^{2} \) |
| 47 | \( 1 - 87.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 5.66iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 62.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 25.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 52.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 90.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 11.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 91.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 42.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 48.1T + 7.92e3T^{2} \) |
| 97 | \( 1 + 5.13iT - 9.40e3T^{2} \) |
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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
−10.48906295184089874012874567381, −9.263116826033988159392062009325, −8.770690236277521968254393858482, −7.84344932336274605427411753010, −7.35544256193224563143696326059, −6.27138041154611978292494233225, −4.29716322133992015392633529872, −3.92696648412474194189360692007, −2.65879253503035908863543076617, −1.75867424163811611972689346213,
1.09840391045799033304774509399, 2.48427369648468529966463057170, 3.49455292838894515767564478921, 4.44019495170889939247695046456, 5.42058293143431188785758629421, 7.38515432431145113382355670919, 7.73013182513436871550445358634, 8.421698738859027213495167138035, 9.204634501993069603016920252113, 9.940618889041909721759000615308