L(s) = 1 | + (0.707 − 0.707i)3-s + (2 + 2i)5-s − 4.24i·7-s − 1.00i·9-s + (−2.82 − 2.82i)11-s + (3 − 3i)13-s + 2.82·15-s − 6·17-s + (1.41 − 1.41i)19-s + (−3 − 3i)21-s + 2.82i·23-s + 3i·25-s + (−0.707 − 0.707i)27-s + (4 − 4i)29-s + 4.24·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (0.894 + 0.894i)5-s − 1.60i·7-s − 0.333i·9-s + (−0.852 − 0.852i)11-s + (0.832 − 0.832i)13-s + 0.730·15-s − 1.45·17-s + (0.324 − 0.324i)19-s + (−0.654 − 0.654i)21-s + 0.589i·23-s + 0.600i·25-s + (−0.136 − 0.136i)27-s + (0.742 − 0.742i)29-s + 0.762·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.58249 - 1.05738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.58249 - 1.05738i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
good | 5 | \( 1 + (-2 - 2i)T + 5iT^{2} \) |
| 7 | \( 1 + 4.24iT - 7T^{2} \) |
| 11 | \( 1 + (2.82 + 2.82i)T + 11iT^{2} \) |
| 13 | \( 1 + (-3 + 3i)T - 13iT^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 + (-1.41 + 1.41i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-4 + 4i)T - 29iT^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 + (-3 - 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 10iT - 41T^{2} \) |
| 43 | \( 1 + (-4.24 - 4.24i)T + 43iT^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + 59iT^{2} \) |
| 61 | \( 1 + (-3 + 3i)T - 61iT^{2} \) |
| 67 | \( 1 + (2.82 - 2.82i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + 4.24T + 79T^{2} \) |
| 83 | \( 1 + (-11.3 + 11.3i)T - 83iT^{2} \) |
| 89 | \( 1 - 14iT - 89T^{2} \) |
| 97 | \( 1 + 4T + 97T^{2} \) |
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
−10.37131985785484597240620743913, −9.440110531263101246493723566702, −8.226697039946861346675248127849, −7.63375413162063429716550619656, −6.55916352975344041183676534129, −6.11445119591430064242779320287, −4.64452540296242581733366135328, −3.37833124370845631094842673982, −2.56414720835923351115198045294, −0.948955856369256371994491761986,
1.90312982309361220561424011402, 2.56869848048385159799366707749, 4.27793153380316927668091195971, 5.13995183399375656247404544564, 5.84178694129719924994683965658, 6.90766429471259305559175216361, 8.440240433763831363181316994940, 8.791547597284304730771994133647, 9.421550462783898529320296037542, 10.26714595602460437713951408608