| L(s) = 1 | + 5·5-s − 7·7-s + 30·11-s − 22·13-s − 48·17-s + 68·19-s − 111·23-s + 25·25-s − 87·29-s + 20·31-s − 35·35-s + 200·37-s − 69·41-s − 232·43-s − 243·47-s − 294·49-s + 498·53-s + 150·55-s − 66·59-s + 359·61-s − 110·65-s − 1.06e3·67-s + 618·71-s − 532·73-s − 210·77-s + 410·79-s − 693·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.822·11-s − 0.469·13-s − 0.684·17-s + 0.821·19-s − 1.00·23-s + 1/5·25-s − 0.557·29-s + 0.115·31-s − 0.169·35-s + 0.888·37-s − 0.262·41-s − 0.822·43-s − 0.754·47-s − 6/7·49-s + 1.29·53-s + 0.367·55-s − 0.145·59-s + 0.753·61-s − 0.209·65-s − 1.93·67-s + 1.03·71-s − 0.852·73-s − 0.310·77-s + 0.583·79-s − 0.916·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - p T \) |
| good | 7 | \( 1 + p T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 + 48 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 111 T + p^{3} T^{2} \) |
| 29 | \( 1 + 3 p T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 200 T + p^{3} T^{2} \) |
| 41 | \( 1 + 69 T + p^{3} T^{2} \) |
| 43 | \( 1 + 232 T + p^{3} T^{2} \) |
| 47 | \( 1 + 243 T + p^{3} T^{2} \) |
| 53 | \( 1 - 498 T + p^{3} T^{2} \) |
| 59 | \( 1 + 66 T + p^{3} T^{2} \) |
| 61 | \( 1 - 359 T + p^{3} T^{2} \) |
| 67 | \( 1 + 1063 T + p^{3} T^{2} \) |
| 71 | \( 1 - 618 T + p^{3} T^{2} \) |
| 73 | \( 1 + 532 T + p^{3} T^{2} \) |
| 79 | \( 1 - 410 T + p^{3} T^{2} \) |
| 83 | \( 1 + 693 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1599 T + p^{3} T^{2} \) |
| 97 | \( 1 - 50 T + p^{3} T^{2} \) |
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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
−8.742589553070066305259619430934, −7.82739887354931146885800851647, −6.91026312593224611052646557762, −6.26835282762337030864542237525, −5.40787808771182573007308992780, −4.42253412982822021917255049841, −3.49690465580331817328929835054, −2.41841258632485958985935098507, −1.37842010135189800399241798750, 0,
1.37842010135189800399241798750, 2.41841258632485958985935098507, 3.49690465580331817328929835054, 4.42253412982822021917255049841, 5.40787808771182573007308992780, 6.26835282762337030864542237525, 6.91026312593224611052646557762, 7.82739887354931146885800851647, 8.742589553070066305259619430934
