| L(s) = 1 | − 2·7-s − 13-s + 17-s + 7·19-s + 8·23-s + 5·29-s + 2·31-s − 7·37-s + 9·41-s + 4·43-s + 47-s − 3·49-s − 9·53-s + 8·59-s − 7·67-s + 5·71-s − 14·73-s + 7·79-s − 18·89-s + 2·91-s − 18·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 2·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.277·13-s + 0.242·17-s + 1.60·19-s + 1.66·23-s + 0.928·29-s + 0.359·31-s − 1.15·37-s + 1.40·41-s + 0.609·43-s + 0.145·47-s − 3/7·49-s − 1.23·53-s + 1.04·59-s − 0.855·67-s + 0.593·71-s − 1.63·73-s + 0.787·79-s − 1.90·89-s + 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.312795748\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.312795748\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.37051293803535, −12.10556186689197, −11.55131942428525, −11.06714025346335, −10.66741471150129, −10.14141829892383, −9.599911121298078, −9.432402843444383, −8.902280336287688, −8.381349988797656, −7.807658834330993, −7.379425381961566, −6.878210769335210, −6.615206104071637, −5.906360371745068, −5.435818566398468, −5.067127775976063, −4.475607031350205, −3.919078056493787, −3.248543997373994, −2.878471653184592, −2.588622035549241, −1.500415875489712, −1.120209317627634, −0.4254611376436830,
0.4254611376436830, 1.120209317627634, 1.500415875489712, 2.588622035549241, 2.878471653184592, 3.248543997373994, 3.919078056493787, 4.475607031350205, 5.067127775976063, 5.435818566398468, 5.906360371745068, 6.615206104071637, 6.878210769335210, 7.379425381961566, 7.807658834330993, 8.381349988797656, 8.902280336287688, 9.432402843444383, 9.599911121298078, 10.14141829892383, 10.66741471150129, 11.06714025346335, 11.55131942428525, 12.10556186689197, 12.37051293803535
