| 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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.864107388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.864107388\) |
| \(L(\frac{3}{2})\) |
|
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
−9.576143999077517550138823183603, −9.169033604206648794010252392969, −8.024242934989618684198899726411, −7.15735240807652252844813757687, −6.36305431055812514656641061202, −5.60045816127901081018108760022, −4.50674267124172289752341210740, −3.55206717706433449462410018262, −2.41907791734996614377815646124, −1.07664647688740391281481003971,
1.07664647688740391281481003971, 2.41907791734996614377815646124, 3.55206717706433449462410018262, 4.50674267124172289752341210740, 5.60045816127901081018108760022, 6.36305431055812514656641061202, 7.15735240807652252844813757687, 8.024242934989618684198899726411, 9.169033604206648794010252392969, 9.576143999077517550138823183603
