L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (−1.86 − 0.5i)13-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.258 + 0.965i)20-s − 1.00i·25-s + 1.93i·26-s + (0.258 − 0.448i)29-s + (−0.965 − 0.258i)32-s + (−1.49 − 0.866i)34-s + (−0.366 − 0.366i)37-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (0.707 − 0.707i)5-s + (0.707 + 0.707i)8-s + (−0.866 − 0.500i)10-s + (−1.86 − 0.5i)13-s + (0.500 − 0.866i)16-s + (1.22 − 1.22i)17-s + (−0.258 + 0.965i)20-s − 1.00i·25-s + 1.93i·26-s + (0.258 − 0.448i)29-s + (−0.965 − 0.258i)32-s + (−1.49 − 0.866i)34-s + (−0.366 − 0.366i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.619 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9142795292\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9142795292\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 + 0.707i)T \) |
good | 7 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.366 + 0.366i)T + iT^{2} \) |
| 41 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 + 0.366i)T - iT^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + 1.93T + T^{2} \) |
| 97 | \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \) |
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
−9.536258371323119232304004031426, −8.812083528672338268737140619345, −7.79410049784833577651000707549, −7.23549380074647655760523003882, −5.64007064082226334073760499375, −5.11827698508651025267893950164, −4.28413145366950377734966843195, −2.93127897965490121460443675320, −2.23196682838475065824164388948, −0.816665320780349375061193051129,
1.66423372131465339649807052837, 2.96442023412964206608957295311, 4.23533263402240901755350994897, 5.20945746278870259014187012133, 5.91326880660120794987454385937, 6.72976824225190660670785213105, 7.40752119965898652786528859510, 8.081758924217614587665030471800, 9.155159563184182508098859232370, 9.824631826796653909743267085934