L(s) = 1 | − 1.41i·2-s − 2.00·4-s + (4.59 + 1.96i)5-s + (−6.81 − 1.57i)7-s + 2.82i·8-s + (2.77 − 6.50i)10-s + 8.15·11-s − 14.6·13-s + (−2.23 + 9.64i)14-s + 4.00·16-s + 5.81·17-s − 33.7i·19-s + (−9.19 − 3.93i)20-s − 11.5i·22-s − 37.2i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.500·4-s + (0.919 + 0.393i)5-s + (−0.974 − 0.225i)7-s + 0.353i·8-s + (0.277 − 0.650i)10-s + 0.741·11-s − 1.12·13-s + (−0.159 + 0.688i)14-s + 0.250·16-s + 0.342·17-s − 1.77i·19-s + (−0.459 − 0.196i)20-s − 0.524i·22-s − 1.61i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.409145721\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.409145721\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.59 - 1.96i)T \) |
| 7 | \( 1 + (6.81 + 1.57i)T \) |
good | 11 | \( 1 - 8.15T + 121T^{2} \) |
| 13 | \( 1 + 14.6T + 169T^{2} \) |
| 17 | \( 1 - 5.81T + 289T^{2} \) |
| 19 | \( 1 + 33.7iT - 361T^{2} \) |
| 23 | \( 1 + 37.2iT - 529T^{2} \) |
| 29 | \( 1 - 9.25T + 841T^{2} \) |
| 31 | \( 1 - 19.2iT - 961T^{2} \) |
| 37 | \( 1 + 63.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 8.25iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 42.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 23.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 71.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 42.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.99iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 124.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 56.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + 90.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 16.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 82.7T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.15697074185624147553459346437, −9.360137738772750028350713437940, −8.810316892003461812890849047565, −7.14180430004940118601763539386, −6.61398163133882498055796888264, −5.42997213977511788322079650181, −4.34665978049061159205377395136, −3.03607337963194695322374748061, −2.26019869947230652803529950760, −0.53107185742560439783213550385,
1.44657849505365206440141840326, 3.05012169359097706381781086409, 4.31686643370082857450318783786, 5.57426905801917317200624205138, 6.05630347857852438152647039165, 7.04423913594777471291499708975, 8.013275744669168199374856547570, 9.102797250420035968519977775457, 9.766657619883268675710651376564, 10.14416246470526000602917015759