L(s) = 1 | + (0.309 + 0.951i)2-s + (−0.809 + 0.587i)4-s + (1.11 − 3.44i)5-s + (3.11 − 2.26i)7-s + (−0.809 − 0.587i)8-s + 3.61·10-s + (−3.23 − 0.726i)11-s + (2 + 6.15i)13-s + (3.11 + 2.26i)14-s + (0.309 − 0.951i)16-s + (−1 + 3.07i)17-s + (−1.61 − 1.17i)19-s + (1.11 + 3.44i)20-s + (−0.309 − 3.30i)22-s + 2.76·23-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.404 + 0.293i)4-s + (0.499 − 1.53i)5-s + (1.17 − 0.856i)7-s + (−0.286 − 0.207i)8-s + 1.14·10-s + (−0.975 − 0.219i)11-s + (0.554 + 1.70i)13-s + (0.833 + 0.605i)14-s + (0.0772 − 0.237i)16-s + (−0.242 + 0.746i)17-s + (−0.371 − 0.269i)19-s + (0.249 + 0.769i)20-s + (−0.0658 − 0.704i)22-s + 0.576·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.43092 + 0.0340250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.43092 + 0.0340250i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (3.23 + 0.726i)T \) |
good | 5 | \( 1 + (-1.11 + 3.44i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-3.11 + 2.26i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-2 - 6.15i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1 - 3.07i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (1.61 + 1.17i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 2.76T + 23T^{2} \) |
| 29 | \( 1 + (0.309 - 0.224i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.0450 - 0.138i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3 - 2.17i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-5.47 - 3.97i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + (3.85 + 2.80i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-2.89 - 8.92i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (9.97 - 7.24i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.854 + 2.62i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + (0.236 - 0.726i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.92 + 1.40i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (2.20 + 6.79i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-2.40 + 7.41i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + (2.20 + 6.79i)T + (-78.4 + 57.0i)T^{2} \) |
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
−12.81711256754345927920658498578, −11.60601211684589455213068036080, −10.54853828869623799796987945448, −9.097836286281403845699204782829, −8.505737784214765755569863843525, −7.50145453927126694687091429115, −6.09986302353272208305694498552, −4.85594093046387365551415191371, −4.31800875702691473113706808648, −1.53576062301919089794728346888,
2.24743704848796746777231755689, 3.13821936892743755580842546584, 5.08747145538125390610482919191, 5.92644019050461093026748684200, 7.46104389249162055933575148093, 8.475433399714909098117673040641, 9.888400903693581023387189339465, 10.80692829784314327378595534410, 11.14756528696380628743561584471, 12.43910069691581540292117441542