L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 5-s − 2·6-s + 7-s + 9-s − 2·10-s − 2·11-s − 2·12-s + 2·14-s + 15-s − 4·16-s + 3·17-s + 2·18-s + 5·19-s − 2·20-s − 21-s − 4·22-s + 23-s + 25-s − 27-s + 2·28-s + 29-s + 2·30-s − 4·31-s − 8·32-s + ⋯ |
L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s + 1/3·9-s − 0.632·10-s − 0.603·11-s − 0.577·12-s + 0.534·14-s + 0.258·15-s − 16-s + 0.727·17-s + 0.471·18-s + 1.14·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 0.185·29-s + 0.365·30-s − 0.718·31-s − 1.41·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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
−16.79025186014036, −15.95677439850830, −15.86243487432184, −15.07921429867695, −14.64669558764262, −13.91419041923949, −13.58219555213783, −12.81758723532614, −12.29740166028256, −11.92043248483400, −11.29287481596666, −10.81540532756453, −10.02410170799300, −9.353127250940880, −8.435229388177375, −7.796780697148437, −7.051750121822555, −6.546307663788831, −5.570797241199663, −5.194549161708319, −4.809737146618708, −3.734191969042389, −3.406857448368338, −2.438025953751510, −1.332242146878808, 0,
1.332242146878808, 2.438025953751510, 3.406857448368338, 3.734191969042389, 4.809737146618708, 5.194549161708319, 5.570797241199663, 6.546307663788831, 7.051750121822555, 7.796780697148437, 8.435229388177375, 9.353127250940880, 10.02410170799300, 10.81540532756453, 11.29287481596666, 11.92043248483400, 12.29740166028256, 12.81758723532614, 13.58219555213783, 13.91419041923949, 14.64669558764262, 15.07921429867695, 15.86243487432184, 15.95677439850830, 16.79025186014036