L(s) = 1 | − 2·2-s + 2·4-s − 2·5-s + 7-s − 3·9-s + 4·10-s + 3·13-s − 2·14-s − 4·16-s − 4·17-s + 6·18-s − 4·19-s − 4·20-s − 9·23-s − 25-s − 6·26-s + 2·28-s + 4·31-s + 8·32-s + 8·34-s − 2·35-s − 6·36-s + 2·37-s + 8·38-s + 2·41-s + 10·43-s + 6·45-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 4-s − 0.894·5-s + 0.377·7-s − 9-s + 1.26·10-s + 0.832·13-s − 0.534·14-s − 16-s − 0.970·17-s + 1.41·18-s − 0.917·19-s − 0.894·20-s − 1.87·23-s − 1/5·25-s − 1.17·26-s + 0.377·28-s + 0.718·31-s + 1.41·32-s + 1.37·34-s − 0.338·35-s − 36-s + 0.328·37-s + 1.29·38-s + 0.312·41-s + 1.52·43-s + 0.894·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 7 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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
−14.18099666364790, −13.72560478968127, −13.43527588614987, −12.50242718075862, −12.12808689405845, −11.47332841370528, −11.25449330118392, −10.68907178133379, −10.48597204957234, −9.583655064247311, −9.230576502273447, −8.680354763191616, −8.229988082425556, −7.926082358962913, −7.626171033886330, −6.743005213410970, −6.188048067132161, −5.944862736089038, −4.883320101437540, −4.300187291871207, −3.985856128571465, −3.111458008196876, −2.357463733711372, −1.860558674138605, −1.059126420480536, 0, 0,
1.059126420480536, 1.860558674138605, 2.357463733711372, 3.111458008196876, 3.985856128571465, 4.300187291871207, 4.883320101437540, 5.944862736089038, 6.188048067132161, 6.743005213410970, 7.626171033886330, 7.926082358962913, 8.229988082425556, 8.680354763191616, 9.230576502273447, 9.583655064247311, 10.48597204957234, 10.68907178133379, 11.25449330118392, 11.47332841370528, 12.12808689405845, 12.50242718075862, 13.43527588614987, 13.72560478968127, 14.18099666364790