L(s) = 1 | + 2-s + 3-s + 4-s + 4·5-s + 6-s + 2·7-s + 8-s + 9-s + 4·10-s − 11-s + 12-s + 2·14-s + 4·15-s + 16-s − 17-s + 18-s + 4·19-s + 4·20-s + 2·21-s − 22-s − 6·23-s + 24-s + 11·25-s + 27-s + 2·28-s + 4·30-s − 10·31-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 1.26·10-s − 0.301·11-s + 0.288·12-s + 0.534·14-s + 1.03·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.894·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 11/5·25-s + 0.192·27-s + 0.377·28-s + 0.730·30-s − 1.79·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−13.39730895900392, −13.19217556593258, −12.42646303114176, −12.14159723197948, −11.54519527103370, −10.88635925937081, −10.47631611909927, −10.13482540456920, −9.548232905397329, −9.186601354482814, −8.574471383184925, −8.175710748974199, −7.457674902018001, −7.051536509370709, −6.536806865130529, −5.887347591457504, −5.464386661813196, −5.147950293083800, −4.620398728322291, −3.860942797040134, −3.296454145318397, −2.762693242867377, −2.051661825818879, −1.702689042770721, −1.390008379515225, 0,
1.390008379515225, 1.702689042770721, 2.051661825818879, 2.762693242867377, 3.296454145318397, 3.860942797040134, 4.620398728322291, 5.147950293083800, 5.464386661813196, 5.887347591457504, 6.536806865130529, 7.051536509370709, 7.457674902018001, 8.175710748974199, 8.574471383184925, 9.186601354482814, 9.548232905397329, 10.13482540456920, 10.47631611909927, 10.88635925937081, 11.54519527103370, 12.14159723197948, 12.42646303114176, 13.19217556593258, 13.39730895900392