L(s) = 1 | − 4·2-s + 10.8·3-s + 16·4-s + 25·5-s − 43.3·6-s + 46.1·7-s − 64·8-s − 125.·9-s − 100·10-s + 8.88·11-s + 173.·12-s + 229.·13-s − 184.·14-s + 270.·15-s + 256·16-s + 561.·17-s + 502.·18-s + 1.95e3·19-s + 400·20-s + 500.·21-s − 35.5·22-s − 529·23-s − 693.·24-s + 625·25-s − 916.·26-s − 3.99e3·27-s + 738.·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.695·3-s + 0.5·4-s + 0.447·5-s − 0.491·6-s + 0.356·7-s − 0.353·8-s − 0.516·9-s − 0.316·10-s + 0.0221·11-s + 0.347·12-s + 0.375·13-s − 0.251·14-s + 0.310·15-s + 0.250·16-s + 0.471·17-s + 0.365·18-s + 1.24·19-s + 0.223·20-s + 0.247·21-s − 0.0156·22-s − 0.208·23-s − 0.245·24-s + 0.200·25-s − 0.265·26-s − 1.05·27-s + 0.178·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.121997401\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.121997401\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
good | 3 | \( 1 - 10.8T + 243T^{2} \) |
| 7 | \( 1 - 46.1T + 1.68e4T^{2} \) |
| 11 | \( 1 - 8.88T + 1.61e5T^{2} \) |
| 13 | \( 1 - 229.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 561.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.95e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 576.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 2.33e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 89.3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.61e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.93e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.74e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.56e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.07e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.76e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.03e5T + 8.58e9T^{2} \) |
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
−11.26859363965375927121406672334, −10.11556571669804639850258671752, −9.345045317540757415694331994507, −8.400446928609162519051753260858, −7.69781340828240996001091270976, −6.37442454010471592772743557374, −5.23032778226310670134154376878, −3.44455361467418303720275773077, −2.32539462411963892627993392093, −0.972401851147545834387009456483,
0.972401851147545834387009456483, 2.32539462411963892627993392093, 3.44455361467418303720275773077, 5.23032778226310670134154376878, 6.37442454010471592772743557374, 7.69781340828240996001091270976, 8.400446928609162519051753260858, 9.345045317540757415694331994507, 10.11556571669804639850258671752, 11.26859363965375927121406672334