L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 3·7-s − 8-s + 9-s − 10-s + 2·11-s − 12-s − 5·13-s + 3·14-s − 15-s + 16-s − 18-s + 6·19-s + 20-s + 3·21-s − 2·22-s + 4·23-s + 24-s + 25-s + 5·26-s − 27-s − 3·28-s − 4·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.603·11-s − 0.288·12-s − 1.38·13-s + 0.801·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.223·20-s + 0.654·21-s − 0.426·22-s + 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.980·26-s − 0.192·27-s − 0.566·28-s − 0.742·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.57234752099703, −12.37909629728020, −11.76032124592406, −11.29578574314233, −10.88814170097100, −10.41669854646663, −9.820624679522490, −9.519233856440377, −9.311802731947975, −8.944591494180156, −8.068354406259983, −7.470315290258784, −7.269344157466848, −6.809756362618759, −6.280169084117223, −5.841650716384226, −5.366792463604699, −4.891113513035005, −4.263291812252188, −3.492155603833494, −3.144752786265067, −2.529388637125389, −1.943740541647342, −1.250223447428676, −0.6555117810828919, 0,
0.6555117810828919, 1.250223447428676, 1.943740541647342, 2.529388637125389, 3.144752786265067, 3.492155603833494, 4.263291812252188, 4.891113513035005, 5.366792463604699, 5.841650716384226, 6.280169084117223, 6.809756362618759, 7.269344157466848, 7.470315290258784, 8.068354406259983, 8.944591494180156, 9.311802731947975, 9.519233856440377, 9.820624679522490, 10.41669854646663, 10.88814170097100, 11.29578574314233, 11.76032124592406, 12.37909629728020, 12.57234752099703