L(s) = 1 | − 2.60·2-s − 3-s + 4.78·4-s + 3.78·5-s + 2.60·6-s − 7-s − 7.26·8-s + 9-s − 9.86·10-s + 2.55·11-s − 4.78·12-s + 2.60·14-s − 3.78·15-s + 9.34·16-s + 3.21·17-s − 2.60·18-s + 8.44·19-s + 18.1·20-s + 21-s − 6.65·22-s − 3.97·23-s + 7.26·24-s + 9.34·25-s − 27-s − 4.78·28-s − 2.86·29-s + 9.86·30-s + ⋯ |
L(s) = 1 | − 1.84·2-s − 0.577·3-s + 2.39·4-s + 1.69·5-s + 1.06·6-s − 0.377·7-s − 2.56·8-s + 0.333·9-s − 3.11·10-s + 0.770·11-s − 1.38·12-s + 0.696·14-s − 0.977·15-s + 2.33·16-s + 0.778·17-s − 0.614·18-s + 1.93·19-s + 4.05·20-s + 0.218·21-s − 1.41·22-s − 0.829·23-s + 1.48·24-s + 1.86·25-s − 0.192·27-s − 0.904·28-s − 0.532·29-s + 1.80·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.025743455\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.025743455\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 - 3.78T + 5T^{2} \) |
| 11 | \( 1 - 2.55T + 11T^{2} \) |
| 17 | \( 1 - 3.21T + 17T^{2} \) |
| 19 | \( 1 - 8.44T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 + 2.86T + 29T^{2} \) |
| 31 | \( 1 + 0.868T + 31T^{2} \) |
| 37 | \( 1 + 5.10T + 37T^{2} \) |
| 41 | \( 1 - 3.34T + 41T^{2} \) |
| 43 | \( 1 + 4.86T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 1.33T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 4.10T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 + 6.76T + 73T^{2} \) |
| 79 | \( 1 + 1.23T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 5.52T + 89T^{2} \) |
| 97 | \( 1 + 5.65T + 97T^{2} \) |
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
−8.939631600315841913317707227952, −7.85315928168469412976919061668, −7.12428566005989063723210421634, −6.52571224657612822728320119823, −5.78384698727643948487145755450, −5.33109748493086213633712959566, −3.56456352650801306577431898593, −2.46798169976891888779596938021, −1.57989423491735853343273940968, −0.877460594827645021565775315814,
0.877460594827645021565775315814, 1.57989423491735853343273940968, 2.46798169976891888779596938021, 3.56456352650801306577431898593, 5.33109748493086213633712959566, 5.78384698727643948487145755450, 6.52571224657612822728320119823, 7.12428566005989063723210421634, 7.85315928168469412976919061668, 8.939631600315841913317707227952