L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 4·19-s − 6·23-s − 5·25-s − 4·26-s − 2·28-s + 6·29-s + 8·31-s − 32-s + 6·34-s − 10·37-s − 4·38-s + 6·41-s − 8·43-s + 6·46-s + 6·47-s − 3·49-s + 5·50-s + 4·52-s + 2·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 1.25·23-s − 25-s − 0.784·26-s − 0.377·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 1.02·34-s − 1.64·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s + 0.884·46-s + 0.875·47-s − 3/7·49-s + 0.707·50-s + 0.554·52-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−8.672134764185654449392944782672, −8.107075144714634957439631011601, −7.11812804222970713232777270208, −6.37459703187143981641083529758, −5.86376231731980296819583852288, −4.54132076139220643267324954006, −3.59300699702492954515692805621, −2.63915482624572011540809421288, −1.45668918986239249375622086896, 0,
1.45668918986239249375622086896, 2.63915482624572011540809421288, 3.59300699702492954515692805621, 4.54132076139220643267324954006, 5.86376231731980296819583852288, 6.37459703187143981641083529758, 7.11812804222970713232777270208, 8.107075144714634957439631011601, 8.672134764185654449392944782672