| L(s) = 1 | + 5-s + 7-s + 3·17-s − 6·19-s − 4·23-s − 4·25-s − 2·29-s + 10·31-s + 35-s − 2·37-s + 10·41-s + 3·43-s − 9·47-s + 49-s + 11·59-s − 5·67-s − 16·73-s + 16·79-s − 9·83-s + 3·85-s − 9·89-s − 6·95-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.727·17-s − 1.37·19-s − 0.834·23-s − 4/5·25-s − 0.371·29-s + 1.79·31-s + 0.169·35-s − 0.328·37-s + 1.56·41-s + 0.457·43-s − 1.31·47-s + 1/7·49-s + 1.43·59-s − 0.610·67-s − 1.87·73-s + 1.80·79-s − 0.987·83-s + 0.325·85-s − 0.953·89-s − 0.615·95-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.45188708396003, −14.20177830124236, −13.39088372356061, −13.22904422928038, −12.44321122775599, −12.09511285458270, −11.48341994246064, −11.05072618222168, −10.25579359276647, −10.11800728386530, −9.486361315008379, −8.870063719168695, −8.250320136132268, −7.939651165050863, −7.308059068376122, −6.563231344715518, −6.087204611093259, −5.670780422805176, −4.960510742610379, −4.271937421445530, −3.928157648658770, −2.986549513667232, −2.361516888610186, −1.779582508754257, −1.008632913036297, 0,
1.008632913036297, 1.779582508754257, 2.361516888610186, 2.986549513667232, 3.928157648658770, 4.271937421445530, 4.960510742610379, 5.670780422805176, 6.087204611093259, 6.563231344715518, 7.308059068376122, 7.939651165050863, 8.250320136132268, 8.870063719168695, 9.486361315008379, 10.11800728386530, 10.25579359276647, 11.05072618222168, 11.48341994246064, 12.09511285458270, 12.44321122775599, 13.22904422928038, 13.39088372356061, 14.20177830124236, 14.45188708396003
