L(s) = 1 | + 2.56i·2-s + 3i·3-s + 1.43·4-s − 7.68·6-s − 25.6i·7-s + 24.1i·8-s − 9·9-s − 11·11-s + 4.31i·12-s + 8.38i·13-s + 65.7·14-s − 50.4·16-s − 8.44i·17-s − 23.0i·18-s − 19.5·19-s + ⋯ |
L(s) = 1 | + 0.905i·2-s + 0.577i·3-s + 0.179·4-s − 0.522·6-s − 1.38i·7-s + 1.06i·8-s − 0.333·9-s − 0.301·11-s + 0.103i·12-s + 0.178i·13-s + 1.25·14-s − 0.787·16-s − 0.120i·17-s − 0.301i·18-s − 0.236·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4273686436\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4273686436\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 3iT \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
good | 2 | \( 1 - 2.56iT - 8T^{2} \) |
| 7 | \( 1 + 25.6iT - 343T^{2} \) |
| 13 | \( 1 - 8.38iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 8.44iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 19.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 72.0iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 375. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 266. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 197. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 594.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 807.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 530. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 85.8T + 3.57e5T^{2} \) |
| 73 | \( 1 - 463. iT - 3.89e5T^{2} \) |
| 79 | \( 1 + 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3iT - 9.12e5T^{2} \) |
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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
−10.40952125126552916261520430709, −9.552969247939184629565857558367, −8.568471300376630852959124456922, −7.48372073170008191596137139719, −7.24705183950514119628526521677, −6.05728263724425500473935935886, −5.24686317579021517241728013238, −4.22402018661211996969484532506, −3.24344785797964173597077755377, −1.68436670441373818851741630742,
0.10045157159132252313517634954, 1.62389679602915740931158045947, 2.45448520418235149526698273852, 3.21549704701045662597581426609, 4.66888728550128065270134003541, 5.88599700646627991236085348184, 6.50929654124474558934257379735, 7.61360152603273073590560658516, 8.549841752173984686301765973624, 9.306432400929326378965712964348