L(s) = 1 | + (0.309 + 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.499 + 1.53i)6-s + (−0.809 − 0.587i)8-s + (0.500 + 1.53i)9-s − 1.61·12-s + (0.309 − 0.951i)16-s + (0.190 − 0.587i)17-s + (−1.30 + 0.951i)18-s + (−0.5 − 0.363i)19-s + (−0.500 − 1.53i)24-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + 32-s + ⋯ |
L(s) = 1 | + (0.309 + 0.951i)2-s + (1.30 + 0.951i)3-s + (−0.809 + 0.587i)4-s + (−0.499 + 1.53i)6-s + (−0.809 − 0.587i)8-s + (0.500 + 1.53i)9-s − 1.61·12-s + (0.309 − 0.951i)16-s + (0.190 − 0.587i)17-s + (−1.30 + 0.951i)18-s + (−0.5 − 0.363i)19-s + (−0.500 − 1.53i)24-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + 32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.573777989\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573777989\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.61T + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 - 0.618T + T^{2} \) |
| 71 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.61T + T^{2} \) |
| 97 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
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
−9.982563868929905173080336031953, −9.579184408579805731547147484890, −8.676031105603438382821176007258, −8.160089558541140703854129973670, −7.31853324634991334469912590615, −6.27584926520502418430643548054, −5.10439875677095502614188382787, −4.32646997337854914351924594432, −3.51122554019010238277258886556, −2.52657730527588627779640910750,
1.48187218211778819161336021793, 2.34124668938990227643161007981, 3.35457367400402607418000321126, 4.14717655794207666675344763851, 5.54712528502313210959620086035, 6.55853461692938904940908181073, 7.68099271914216795176554973565, 8.352524705634098925172614246273, 9.100638984589307542928765534911, 9.834691257733438615757864561538