| L(s) = 1 | + (0.383 − 2.88i)3-s + (1.71 − 0.758i)5-s + (0.166 + 0.358i)7-s + (−5.25 − 1.42i)9-s + (0.801 + 1.02i)11-s + (−2.57 − 5.03i)13-s + (−1.52 − 5.23i)15-s + (1.81 − 0.137i)17-s + (−1.54 − 2.32i)19-s + (1.09 − 0.342i)21-s + (4.00 − 1.59i)23-s + (−0.999 + 1.09i)25-s + (−2.75 + 6.55i)27-s + (−3.15 − 1.25i)29-s + (4.27 + 0.988i)31-s + ⋯ |
| L(s) = 1 | + (0.221 − 1.66i)3-s + (0.766 − 0.339i)5-s + (0.0630 + 0.135i)7-s + (−1.75 − 0.475i)9-s + (0.241 + 0.309i)11-s + (−0.715 − 1.39i)13-s + (−0.393 − 1.35i)15-s + (0.439 − 0.0333i)17-s + (−0.354 − 0.533i)19-s + (0.239 − 0.0748i)21-s + (0.835 − 0.332i)23-s + (−0.199 + 0.219i)25-s + (−0.529 + 1.26i)27-s + (−0.585 − 0.232i)29-s + (0.767 + 0.177i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 668 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.694 + 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.655643 - 1.54422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.655643 - 1.54422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 167 | \( 1 + (-12.7 - 2.16i)T \) |
| good | 3 | \( 1 + (-0.383 + 2.88i)T + (-2.89 - 0.785i)T^{2} \) |
| 5 | \( 1 + (-1.71 + 0.758i)T + (3.36 - 3.69i)T^{2} \) |
| 7 | \( 1 + (-0.166 - 0.358i)T + (-4.51 + 5.35i)T^{2} \) |
| 11 | \( 1 + (-0.801 - 1.02i)T + (-2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (2.57 + 5.03i)T + (-7.60 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.81 + 0.137i)T + (16.8 - 2.56i)T^{2} \) |
| 19 | \( 1 + (1.54 + 2.32i)T + (-7.35 + 17.5i)T^{2} \) |
| 23 | \( 1 + (-4.00 + 1.59i)T + (16.7 - 15.7i)T^{2} \) |
| 29 | \( 1 + (3.15 + 1.25i)T + (21.0 + 19.9i)T^{2} \) |
| 31 | \( 1 + (-4.27 - 0.988i)T + (27.8 + 13.6i)T^{2} \) |
| 37 | \( 1 + (2.37 - 0.645i)T + (31.9 - 18.7i)T^{2} \) |
| 41 | \( 1 + (0.773 + 0.377i)T + (25.2 + 32.3i)T^{2} \) |
| 43 | \( 1 + (-0.670 - 7.06i)T + (-42.2 + 8.08i)T^{2} \) |
| 47 | \( 1 + (0.216 - 11.4i)T + (-46.9 - 1.77i)T^{2} \) |
| 53 | \( 1 + (-1.11 + 6.49i)T + (-49.9 - 17.7i)T^{2} \) |
| 59 | \( 1 + (-1.61 - 0.122i)T + (58.3 + 8.89i)T^{2} \) |
| 61 | \( 1 + (4.42 + 4.50i)T + (-1.15 + 60.9i)T^{2} \) |
| 67 | \( 1 + (1.47 + 0.650i)T + (45.0 + 49.5i)T^{2} \) |
| 71 | \( 1 + (-4.98 - 0.568i)T + (69.1 + 15.9i)T^{2} \) |
| 73 | \( 1 + (-6.50 - 4.87i)T + (20.4 + 70.0i)T^{2} \) |
| 79 | \( 1 + (-4.64 - 12.3i)T + (-59.4 + 52.0i)T^{2} \) |
| 83 | \( 1 + (4.25 + 5.90i)T + (-26.2 + 78.7i)T^{2} \) |
| 89 | \( 1 + (-1.27 + 4.35i)T + (-75.0 - 47.8i)T^{2} \) |
| 97 | \( 1 + (-11.7 + 2.71i)T + (87.1 - 42.5i)T^{2} \) |
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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
−10.08331196622719221913127345870, −9.211301247117296301043898457952, −8.245741980764478440651364586129, −7.56871619550304131250242115397, −6.71256561641422525871110902438, −5.84197650940812983166761614662, −4.97202690273980091724696838599, −3.02959881244510076539535799671, −2.09629459505106474151065483217, −0.891501559639468665019803004335,
2.15770283934883453519833276334, 3.43666246120488227575244540968, 4.31873944924784372782029670682, 5.23301904521008017843541144401, 6.17457370199233637625292793410, 7.30108891079803358757449054203, 8.685969645352564361524885539039, 9.261005117439279993298626522851, 10.00281552237180316380041661904, 10.52753500233549996718760383503
