L(s) = 1 | + 2-s + 4-s + 8-s − 4·11-s − 3·13-s + 16-s + 7·17-s + 6·19-s − 4·22-s − 9·23-s − 3·26-s − 3·29-s + 7·31-s + 32-s + 7·34-s + 10·37-s + 6·38-s − 41-s − 13·43-s − 4·44-s − 9·46-s − 2·47-s − 3·52-s + 53-s − 3·58-s − 11·59-s − 13·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 1.20·11-s − 0.832·13-s + 1/4·16-s + 1.69·17-s + 1.37·19-s − 0.852·22-s − 1.87·23-s − 0.588·26-s − 0.557·29-s + 1.25·31-s + 0.176·32-s + 1.20·34-s + 1.64·37-s + 0.973·38-s − 0.156·41-s − 1.98·43-s − 0.603·44-s − 1.32·46-s − 0.291·47-s − 0.416·52-s + 0.137·53-s − 0.393·58-s − 1.43·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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
−15.78406300923862, −15.11583043547603, −14.79277790298533, −14.03055205435790, −13.71757193241845, −13.21663015694261, −12.41113612852554, −12.05725433860009, −11.72788544344291, −10.91222234077137, −10.18825645438457, −9.874720287450342, −9.442922515191579, −8.153829656888899, −7.855661248136415, −7.554303356054447, −6.645142849708695, −5.892249592845003, −5.486208218366316, −4.880743954617296, −4.274070727123961, −3.266509399899083, −2.979010111763404, −2.105610102352209, −1.212935408209919, 0,
1.212935408209919, 2.105610102352209, 2.979010111763404, 3.266509399899083, 4.274070727123961, 4.880743954617296, 5.486208218366316, 5.892249592845003, 6.645142849708695, 7.554303356054447, 7.855661248136415, 8.153829656888899, 9.442922515191579, 9.874720287450342, 10.18825645438457, 10.91222234077137, 11.72788544344291, 12.05725433860009, 12.41113612852554, 13.21663015694261, 13.71757193241845, 14.03055205435790, 14.79277790298533, 15.11583043547603, 15.78406300923862