L(s) = 1 | + 2-s − 3-s + 4-s + 3·5-s − 6-s − 3·7-s + 8-s + 9-s + 3·10-s + 11-s − 12-s + 13-s − 3·14-s − 3·15-s + 16-s + 3·17-s + 18-s − 4·19-s + 3·20-s + 3·21-s + 22-s − 24-s + 4·25-s + 26-s − 27-s − 3·28-s + 7·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.34·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s − 0.774·15-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.917·19-s + 0.670·20-s + 0.654·21-s + 0.213·22-s − 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.192·27-s − 0.566·28-s + 1.29·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34914 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.482202085\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.482202085\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.90638240621773, −14.14832053300829, −13.96525912255748, −13.30088867034288, −12.90055399444315, −12.40531766298159, −12.05579880999425, −11.23615312803166, −10.65009997673673, −10.20502379631941, −9.762982211373318, −9.257343778442214, −8.554482118085481, −7.824796256490638, −6.864394056719168, −6.501897523963035, −6.260980035932600, −5.519182202734350, −5.140740131820270, −4.372010083933735, −3.538214833942487, −3.101757802874962, −2.173144772708060, −1.627440989027398, −0.6303215889389511,
0.6303215889389511, 1.627440989027398, 2.173144772708060, 3.101757802874962, 3.538214833942487, 4.372010083933735, 5.140740131820270, 5.519182202734350, 6.260980035932600, 6.501897523963035, 6.864394056719168, 7.824796256490638, 8.554482118085481, 9.257343778442214, 9.762982211373318, 10.20502379631941, 10.65009997673673, 11.23615312803166, 12.05579880999425, 12.40531766298159, 12.90055399444315, 13.30088867034288, 13.96525912255748, 14.14832053300829, 14.90638240621773