| L(s) = 1 | + 7-s − 3·11-s + 13-s + 7·17-s + 4·19-s + 9·23-s − 8·29-s − 4·31-s − 3·37-s + 5·41-s + 2·47-s − 6·49-s + 3·53-s − 12·59-s − 15·61-s + 12·67-s − 15·71-s + 2·73-s − 3·77-s + 5·79-s − 10·83-s − 5·89-s + 91-s − 13·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s + 1.69·17-s + 0.917·19-s + 1.87·23-s − 1.48·29-s − 0.718·31-s − 0.493·37-s + 0.780·41-s + 0.291·47-s − 6/7·49-s + 0.412·53-s − 1.56·59-s − 1.92·61-s + 1.46·67-s − 1.78·71-s + 0.234·73-s − 0.341·77-s + 0.562·79-s − 1.09·83-s − 0.529·89-s + 0.104·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.77146686793565, −14.42336763631451, −13.84111806686275, −13.28284801709828, −12.79986511309373, −12.36725122665214, −11.75489582152723, −11.08134372591704, −10.85495191017021, −10.24007415260876, −9.484651788814806, −9.268601771344448, −8.503069858207906, −7.863011385751352, −7.424660078834542, −7.146984121022029, −6.109409077890353, −5.559793330121234, −5.206109240996404, −4.623387140951064, −3.673298988763767, −3.192797265269079, −2.652645986745035, −1.578285006694344, −1.121395928592429, 0,
1.121395928592429, 1.578285006694344, 2.652645986745035, 3.192797265269079, 3.673298988763767, 4.623387140951064, 5.206109240996404, 5.559793330121234, 6.109409077890353, 7.146984121022029, 7.424660078834542, 7.863011385751352, 8.503069858207906, 9.268601771344448, 9.484651788814806, 10.24007415260876, 10.85495191017021, 11.08134372591704, 11.75489582152723, 12.36725122665214, 12.79986511309373, 13.28284801709828, 13.84111806686275, 14.42336763631451, 14.77146686793565
