L(s) = 1 | + 3-s − 2·4-s − 3·5-s − 2·9-s − 2·12-s − 3·15-s + 4·16-s + 6·20-s + 9·23-s + 4·25-s − 5·27-s + 4·36-s − 7·37-s + 6·45-s − 12·47-s + 4·48-s − 7·49-s − 6·53-s − 15·59-s + 6·60-s − 8·64-s + 13·67-s + 9·69-s − 3·71-s + 4·75-s − 12·80-s + 81-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 1.34·5-s − 2/3·9-s − 0.577·12-s − 0.774·15-s + 16-s + 1.34·20-s + 1.87·23-s + 4/5·25-s − 0.962·27-s + 2/3·36-s − 1.15·37-s + 0.894·45-s − 1.75·47-s + 0.577·48-s − 49-s − 0.824·53-s − 1.95·59-s + 0.774·60-s − 64-s + 1.58·67-s + 1.08·69-s − 0.356·71-s + 0.461·75-s − 1.34·80-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5201313905\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5201313905\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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
−13.59645932004375, −13.13429864980739, −12.67917597085912, −12.23202388278913, −11.66120726307688, −11.08855209796937, −10.93010360653516, −10.04709076216472, −9.563753995861224, −9.029050308509490, −8.616590357824690, −8.309331070682500, −7.676073091091474, −7.450132641791327, −6.625804770001535, −6.102447449911347, −5.184756444580701, −4.923195353645209, −4.426225847326836, −3.570139007951366, −3.367313608065065, −2.949656891849389, −1.915322139150954, −1.080384389334288, −0.2459114833353112,
0.2459114833353112, 1.080384389334288, 1.915322139150954, 2.949656891849389, 3.367313608065065, 3.570139007951366, 4.426225847326836, 4.923195353645209, 5.184756444580701, 6.102447449911347, 6.625804770001535, 7.450132641791327, 7.676073091091474, 8.309331070682500, 8.616590357824690, 9.029050308509490, 9.563753995861224, 10.04709076216472, 10.93010360653516, 11.08855209796937, 11.66120726307688, 12.23202388278913, 12.67917597085912, 13.13429864980739, 13.59645932004375