L(s) = 1 | − 1.61·2-s − 1.01·3-s + 0.612·4-s + 4.09·5-s + 1.63·6-s − 3.10·7-s + 2.24·8-s − 1.97·9-s − 6.61·10-s − 11-s − 0.621·12-s − 4.38·13-s + 5.01·14-s − 4.14·15-s − 4.85·16-s − 2.31·17-s + 3.18·18-s + 7.39·19-s + 2.50·20-s + 3.14·21-s + 1.61·22-s − 1.81·23-s − 2.27·24-s + 11.7·25-s + 7.08·26-s + 5.04·27-s − 1.89·28-s + ⋯ |
L(s) = 1 | − 1.14·2-s − 0.585·3-s + 0.306·4-s + 1.83·5-s + 0.668·6-s − 1.17·7-s + 0.792·8-s − 0.657·9-s − 2.09·10-s − 0.301·11-s − 0.179·12-s − 1.21·13-s + 1.33·14-s − 1.07·15-s − 1.21·16-s − 0.560·17-s + 0.751·18-s + 1.69·19-s + 0.560·20-s + 0.685·21-s + 0.344·22-s − 0.378·23-s − 0.463·24-s + 2.35·25-s + 1.39·26-s + 0.969·27-s − 0.358·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6576510071\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6576510071\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 - 4.09T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 13 | \( 1 + 4.38T + 13T^{2} \) |
| 17 | \( 1 + 2.31T + 17T^{2} \) |
| 19 | \( 1 - 7.39T + 19T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 9.07T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 0.113T + 41T^{2} \) |
| 43 | \( 1 + 0.277T + 43T^{2} \) |
| 47 | \( 1 + 1.35T + 47T^{2} \) |
| 53 | \( 1 + 2.17T + 53T^{2} \) |
| 59 | \( 1 - 1.50T + 59T^{2} \) |
| 61 | \( 1 - 3.32T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 5.49T + 73T^{2} \) |
| 79 | \( 1 + 5.98T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 - 6.54T + 97T^{2} \) |
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
−9.622520910710937778406009766374, −9.111845288208408742803009214578, −8.100002596116647845798500207592, −6.96834752469461103793545348036, −6.40741753630123717979483262431, −5.47082058261216323077683089659, −4.89836789034801633061475071164, −3.00652518461785122003669460429, −2.14011629895950778257720148749, −0.69360951520977806801534151486,
0.69360951520977806801534151486, 2.14011629895950778257720148749, 3.00652518461785122003669460429, 4.89836789034801633061475071164, 5.47082058261216323077683089659, 6.40741753630123717979483262431, 6.96834752469461103793545348036, 8.100002596116647845798500207592, 9.111845288208408742803009214578, 9.622520910710937778406009766374