| L(s) = 1 | − 5-s − 7-s − 5·11-s + 3·13-s + 17-s − 6·19-s − 6·23-s + 25-s − 9·29-s − 4·31-s + 35-s − 2·37-s + 4·41-s − 10·43-s + 47-s + 49-s + 4·53-s + 5·55-s − 8·59-s + 8·61-s − 3·65-s − 12·67-s − 8·71-s + 2·73-s + 5·77-s + 13·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.50·11-s + 0.832·13-s + 0.242·17-s − 1.37·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s − 0.718·31-s + 0.169·35-s − 0.328·37-s + 0.624·41-s − 1.52·43-s + 0.145·47-s + 1/7·49-s + 0.549·53-s + 0.674·55-s − 1.04·59-s + 1.02·61-s − 0.372·65-s − 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.569·77-s + 1.46·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4393743249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4393743249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−15.71079899451327, −15.07720218632284, −14.79447420993253, −13.92310515246531, −13.31773860345845, −12.99475942544506, −12.48358258068861, −11.80525372932493, −11.16792709095347, −10.60111102661343, −10.30540161544432, −9.521515586304485, −8.841534276590902, −8.268412077766571, −7.789866572697960, −7.227658339172502, −6.433528848296475, −5.817095147395290, −5.333345928872020, −4.436185629282083, −3.823111488974195, −3.233474721386558, −2.336517117576303, −1.675683178001127, −0.2593215966966254,
0.2593215966966254, 1.675683178001127, 2.336517117576303, 3.233474721386558, 3.823111488974195, 4.436185629282083, 5.333345928872020, 5.817095147395290, 6.433528848296475, 7.227658339172502, 7.789866572697960, 8.268412077766571, 8.841534276590902, 9.521515586304485, 10.30540161544432, 10.60111102661343, 11.16792709095347, 11.80525372932493, 12.48358258068861, 12.99475942544506, 13.31773860345845, 13.92310515246531, 14.79447420993253, 15.07720218632284, 15.71079899451327
