L(s) = 1 | − 2·5-s + 7-s − 3·9-s − 11-s − 2·13-s + 2·17-s + 8·23-s − 25-s + 2·29-s + 8·31-s − 2·35-s + 2·37-s + 10·41-s + 4·43-s + 6·45-s − 8·47-s + 49-s − 6·53-s + 2·55-s − 10·61-s − 3·63-s + 4·65-s − 12·67-s − 16·71-s − 14·73-s − 77-s + 9·81-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.377·7-s − 9-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.338·35-s + 0.328·37-s + 1.56·41-s + 0.609·43-s + 0.894·45-s − 1.16·47-s + 1/7·49-s − 0.824·53-s + 0.269·55-s − 1.28·61-s − 0.377·63-s + 0.496·65-s − 1.46·67-s − 1.89·71-s − 1.63·73-s − 0.113·77-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−7.72024110024881070821541339163, −7.52779608421941576346985997734, −6.42169131751856764457567127082, −5.69715804540238922198121355930, −4.82761578982425423372007838462, −4.30916127979318708251021123277, −3.08172507755908328232950357793, −2.72854508523448958533903853214, −1.21399850656019313589842252973, 0,
1.21399850656019313589842252973, 2.72854508523448958533903853214, 3.08172507755908328232950357793, 4.30916127979318708251021123277, 4.82761578982425423372007838462, 5.69715804540238922198121355930, 6.42169131751856764457567127082, 7.52779608421941576346985997734, 7.72024110024881070821541339163