| L(s) = 1 | − 7-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s − 23-s + 2·29-s + 8·31-s − 6·37-s + 6·41-s + 4·43-s + 8·47-s + 49-s − 10·53-s + 12·59-s + 14·61-s + 12·67-s − 16·71-s + 14·73-s − 4·77-s + 8·79-s − 12·83-s + 14·89-s − 2·91-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 0.208·23-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s + 1.56·59-s + 1.79·61-s + 1.46·67-s − 1.89·71-s + 1.63·73-s − 0.455·77-s + 0.900·79-s − 1.31·83-s + 1.48·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.91842339943124, −12.45237742115623, −12.03561191253665, −11.64414315117296, −11.14831132724318, −10.58336739234491, −10.27670907880452, −9.630118382824937, −9.337306015571620, −8.774875747777251, −8.315223492132235, −8.002435844116424, −7.230084527137819, −6.709564270139532, −6.480507621376269, −5.929113435338354, −5.452558375055819, −4.778516059772327, −4.179087442193471, −3.806918455860903, −3.394505186508651, −2.501263028556129, −2.243262841263503, −1.163299224748208, −1.020851132250599, 0,
1.020851132250599, 1.163299224748208, 2.243262841263503, 2.501263028556129, 3.394505186508651, 3.806918455860903, 4.179087442193471, 4.778516059772327, 5.452558375055819, 5.929113435338354, 6.480507621376269, 6.709564270139532, 7.230084527137819, 8.002435844116424, 8.315223492132235, 8.774875747777251, 9.337306015571620, 9.630118382824937, 10.27670907880452, 10.58336739234491, 11.14831132724318, 11.64414315117296, 12.03561191253665, 12.45237742115623, 12.91842339943124
