| L(s) = 1 | − 5-s − 4·7-s − 3·9-s + 4·11-s − 13-s + 2·17-s − 4·19-s + 25-s − 6·29-s − 4·31-s + 4·35-s − 6·37-s + 10·41-s − 8·43-s + 3·45-s + 12·47-s + 9·49-s + 10·53-s − 4·55-s − 12·59-s − 6·61-s + 12·63-s + 65-s − 4·67-s − 4·71-s + 2·73-s − 16·77-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.51·7-s − 9-s + 1.20·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.676·35-s − 0.986·37-s + 1.56·41-s − 1.21·43-s + 0.447·45-s + 1.75·47-s + 9/7·49-s + 1.37·53-s − 0.539·55-s − 1.56·59-s − 0.768·61-s + 1.51·63-s + 0.124·65-s − 0.488·67-s − 0.474·71-s + 0.234·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8798853626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8798853626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−8.679522631945850602135816308926, −7.51342069568065680500641069103, −6.99779353343744035287305408594, −6.07339582046800137758081173074, −5.78648153062773281226115590413, −4.49918355206622824994104171942, −3.63530767753440735417837464328, −3.17651296087682402103225680541, −2.05941271660251934670619371489, −0.50759725669808292974043415692,
0.50759725669808292974043415692, 2.05941271660251934670619371489, 3.17651296087682402103225680541, 3.63530767753440735417837464328, 4.49918355206622824994104171942, 5.78648153062773281226115590413, 6.07339582046800137758081173074, 6.99779353343744035287305408594, 7.51342069568065680500641069103, 8.679522631945850602135816308926
