| 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
−10.14416246470526000602917015759, −9.766657619883268675710651376564, −9.102797250420035968519977775457, −8.013275744669168199374856547570, −7.04423913594777471291499708975, −6.05630347857852438152647039165, −5.57426905801917317200624205138, −4.31686643370082857450318783786, −3.05012169359097706381781086409, −1.44657849505365206440141840326,
0.53107185742560439783213550385, 2.26019869947230652803529950760, 3.03607337963194695322374748061, 4.34665978049061159205377395136, 5.42997213977511788322079650181, 6.61398163133882498055796888264, 7.14180430004940118601763539386, 8.810316892003461812890849047565, 9.360137738772750028350713437940, 10.15697074185624147553459346437
