L(s) = 1 | + (−1.37 − 0.315i)2-s + (0.987 + 0.156i)3-s + (1.80 + 0.869i)4-s + (2.22 + 0.203i)5-s + (−1.31 − 0.527i)6-s + (2.91 − 2.91i)7-s + (−2.20 − 1.76i)8-s + (0.951 + 0.309i)9-s + (−3.00 − 0.982i)10-s + (−2.79 + 0.909i)11-s + (1.64 + 1.14i)12-s + (−0.922 − 1.81i)13-s + (−4.93 + 3.09i)14-s + (2.16 + 0.549i)15-s + (2.48 + 3.13i)16-s + (0.653 + 4.12i)17-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.223i)2-s + (0.570 + 0.0903i)3-s + (0.900 + 0.434i)4-s + (0.995 + 0.0909i)5-s + (−0.535 − 0.215i)6-s + (1.10 − 1.10i)7-s + (−0.780 − 0.624i)8-s + (0.317 + 0.103i)9-s + (−0.950 − 0.310i)10-s + (−0.843 + 0.274i)11-s + (0.474 + 0.329i)12-s + (−0.255 − 0.502i)13-s + (−1.31 + 0.827i)14-s + (0.559 + 0.141i)15-s + (0.621 + 0.783i)16-s + (0.158 + 1.00i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.916 + 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.23919 - 0.258948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23919 - 0.258948i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 + 0.315i)T \) |
| 3 | \( 1 + (-0.987 - 0.156i)T \) |
| 5 | \( 1 + (-2.22 - 0.203i)T \) |
good | 7 | \( 1 + (-2.91 + 2.91i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.79 - 0.909i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (0.922 + 1.81i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.653 - 4.12i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (2.04 + 1.48i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.577 + 1.13i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (4.10 + 5.65i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (4.56 - 6.28i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-9.09 + 4.63i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (1.48 - 4.56i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-4.56 - 4.56i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.753 - 4.75i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.396 + 2.50i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (1.41 - 4.34i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.01 - 9.27i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (14.9 - 2.36i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (-3.33 - 4.58i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (7.97 + 4.06i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (7.70 - 5.60i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.36 + 14.9i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-3.68 + 1.19i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (17.8 + 2.82i)T + (92.2 + 29.9i)T^{2} \) |
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
−11.18210605768073799380507211401, −10.52677817506802325132598848344, −9.959258442970377910318474067880, −8.853489326115811203883180039710, −7.896433213968866922875249502736, −7.28149196353799492594823500260, −5.87124870674345445273034545202, −4.35733571873914228593724408157, −2.69655203418185935459627741100, −1.50408334670466798406493683199,
1.81606470498647717376982698914, 2.63788817332231409692386699278, 5.10308904754891714813269816571, 5.87097573608176853060357344194, 7.24405554601224017313398256974, 8.165206147775521640883068361151, 9.016013592476608098302717110369, 9.582811380376105501641248233038, 10.72865789151859422095288162462, 11.57101963970000836515702076885