L(s) = 1 | + 2-s − 0.922·3-s + 4-s + 5-s − 0.922·6-s − 2.45·7-s + 8-s − 2.14·9-s + 10-s − 2.04·11-s − 0.922·12-s + 2.73·13-s − 2.45·14-s − 0.922·15-s + 16-s − 1.97·17-s − 2.14·18-s + 2.65·19-s + 20-s + 2.26·21-s − 2.04·22-s + 3.70·23-s − 0.922·24-s + 25-s + 2.73·26-s + 4.75·27-s − 2.45·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.532·3-s + 0.5·4-s + 0.447·5-s − 0.376·6-s − 0.926·7-s + 0.353·8-s − 0.716·9-s + 0.316·10-s − 0.617·11-s − 0.266·12-s + 0.758·13-s − 0.655·14-s − 0.238·15-s + 0.250·16-s − 0.478·17-s − 0.506·18-s + 0.609·19-s + 0.223·20-s + 0.493·21-s − 0.436·22-s + 0.773·23-s − 0.188·24-s + 0.200·25-s + 0.536·26-s + 0.914·27-s − 0.463·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 3 | \( 1 + 0.922T + 3T^{2} \) |
| 7 | \( 1 + 2.45T + 7T^{2} \) |
| 11 | \( 1 + 2.04T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 1.97T + 17T^{2} \) |
| 19 | \( 1 - 2.65T + 19T^{2} \) |
| 23 | \( 1 - 3.70T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 5.51T + 31T^{2} \) |
| 37 | \( 1 + 0.227T + 37T^{2} \) |
| 41 | \( 1 - 0.00373T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 + 7.35T + 47T^{2} \) |
| 53 | \( 1 + 4.94T + 53T^{2} \) |
| 59 | \( 1 - 0.966T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 2.64T + 67T^{2} \) |
| 71 | \( 1 + 0.355T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 15.2T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 + 17.8T + 89T^{2} \) |
| 97 | \( 1 + 12.0T + 97T^{2} \) |
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
−8.056044827928252697680215604871, −7.00363594877895947460495912999, −6.43461541696894871454976517561, −5.81285780694750098457453941335, −5.24061200148550478001782643496, −4.40294815007705690897953523318, −3.18959234610140552352245790644, −2.86951895118045451424872676434, −1.49578716787361820520992146960, 0,
1.49578716787361820520992146960, 2.86951895118045451424872676434, 3.18959234610140552352245790644, 4.40294815007705690897953523318, 5.24061200148550478001782643496, 5.81285780694750098457453941335, 6.43461541696894871454976517561, 7.00363594877895947460495912999, 8.056044827928252697680215604871