L(s) = 1 | + 1.32·2-s − 6.25·4-s − 12.9·5-s + 5.54·7-s − 18.8·8-s − 17.1·10-s + 0.841·11-s + 7.33·14-s + 25.1·16-s − 10.1·17-s − 73.0·19-s + 81.0·20-s + 1.11·22-s − 139.·23-s + 42.9·25-s − 34.6·28-s − 250.·29-s − 161.·31-s + 183.·32-s − 13.4·34-s − 71.8·35-s + 195.·37-s − 96.6·38-s + 244.·40-s − 183.·41-s + 115.·43-s − 5.26·44-s + ⋯ |
L(s) = 1 | + 0.467·2-s − 0.781·4-s − 1.15·5-s + 0.299·7-s − 0.832·8-s − 0.541·10-s + 0.0230·11-s + 0.139·14-s + 0.392·16-s − 0.145·17-s − 0.882·19-s + 0.905·20-s + 0.0107·22-s − 1.26·23-s + 0.343·25-s − 0.233·28-s − 1.60·29-s − 0.935·31-s + 1.01·32-s − 0.0679·34-s − 0.347·35-s + 0.870·37-s − 0.412·38-s + 0.965·40-s − 0.699·41-s + 0.408·43-s − 0.0180·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.6767174699\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6767174699\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.32T + 8T^{2} \) |
| 5 | \( 1 + 12.9T + 125T^{2} \) |
| 7 | \( 1 - 5.54T + 343T^{2} \) |
| 11 | \( 1 - 0.841T + 1.33e3T^{2} \) |
| 17 | \( 1 + 10.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 139.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 250.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 161.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 115.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 551.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 324.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 521.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 444.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 55.0T + 3.00e5T^{2} \) |
| 71 | \( 1 + 279.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 908.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 941.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 946.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 32.9T + 7.04e5T^{2} \) |
| 97 | \( 1 + 68.2T + 9.12e5T^{2} \) |
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
−9.039885351088388523222061726318, −8.135802860886362446506803310306, −7.77826868755451145629418782348, −6.59564757721817727338774168571, −5.65833030009647955484712346657, −4.76210285852855417730072372599, −3.97945388578198647806786129711, −3.48865662600628674599406610096, −1.99526564013607588434688767928, −0.36282017914891361375980738579,
0.36282017914891361375980738579, 1.99526564013607588434688767928, 3.48865662600628674599406610096, 3.97945388578198647806786129711, 4.76210285852855417730072372599, 5.65833030009647955484712346657, 6.59564757721817727338774168571, 7.77826868755451145629418782348, 8.135802860886362446506803310306, 9.039885351088388523222061726318