L(s) = 1 | + (−1.55 − 1.26i)2-s + (2.10 − 2.10i)3-s + (0.813 + 3.91i)4-s + (−4.62 + 4.62i)5-s + (−5.91 + 0.608i)6-s + 3.04·7-s + (3.68 − 7.10i)8-s + 0.156i·9-s + (13.0 − 1.33i)10-s + (−9.15 − 9.15i)11-s + (9.94 + 6.52i)12-s + (−5.78 − 5.78i)13-s + (−4.72 − 3.84i)14-s + 19.4i·15-s + (−14.6 + 6.37i)16-s + 17.6·17-s + ⋯ |
L(s) = 1 | + (−0.775 − 0.631i)2-s + (0.700 − 0.700i)3-s + (0.203 + 0.979i)4-s + (−0.925 + 0.925i)5-s + (−0.986 + 0.101i)6-s + 0.435·7-s + (0.460 − 0.887i)8-s + 0.0174i·9-s + (1.30 − 0.133i)10-s + (−0.831 − 0.831i)11-s + (0.828 + 0.543i)12-s + (−0.444 − 0.444i)13-s + (−0.337 − 0.274i)14-s + 1.29i·15-s + (−0.917 + 0.398i)16-s + 1.03·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.595321 - 0.240695i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.595321 - 0.240695i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.55 + 1.26i)T \) |
good | 3 | \( 1 + (-2.10 + 2.10i)T - 9iT^{2} \) |
| 5 | \( 1 + (4.62 - 4.62i)T - 25iT^{2} \) |
| 7 | \( 1 - 3.04T + 49T^{2} \) |
| 11 | \( 1 + (9.15 + 9.15i)T + 121iT^{2} \) |
| 13 | \( 1 + (5.78 + 5.78i)T + 169iT^{2} \) |
| 17 | \( 1 - 17.6T + 289T^{2} \) |
| 19 | \( 1 + (1.15 - 1.15i)T - 361iT^{2} \) |
| 23 | \( 1 + 3.45T + 529T^{2} \) |
| 29 | \( 1 + (-12.1 - 12.1i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.5iT - 961T^{2} \) |
| 37 | \( 1 + (0.0972 - 0.0972i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 51.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (1.70 + 1.70i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 24.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-27.0 + 27.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-19.5 - 19.5i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-16.7 - 16.7i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (75.8 - 75.8i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 134.T + 5.04e3T^{2} \) |
| 73 | \( 1 + 112. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 135. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-74.9 + 74.9i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 31.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 31.5T + 9.40e3T^{2} \) |
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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
−18.93018841176314664890974775170, −18.15104896815027068919475755278, −16.35332775477675652668389363409, −14.76304392848751706692529374921, −13.21334748146507348543615789638, −11.66918127289927349514823795935, −10.43946517157300583994667948221, −8.200020929716778120328955933920, −7.49795545202933616860365622989, −2.98931204697392862733797842527,
4.75617402699940195329251308925, 7.66736938703655029869243953012, 8.811955731886072562243428547535, 10.14434279059121646655929681001, 12.12711696639815969211762356962, 14.38367411776500791876881749275, 15.44155438872578373344285833964, 16.27321399969846010193682678422, 17.69273754984693331912412168442, 19.24631576833940179260032195605