L(s) = 1 | + (−1.22 − 0.707i)2-s + (0.999 + 1.73i)4-s + (−5.25 − 3.03i)5-s − 2.82i·8-s + (4.29 + 7.43i)10-s + (10.5 − 6.06i)11-s − 18.5·13-s + (−2.00 + 3.46i)16-s + (9.44 − 5.45i)17-s + (−10 + 17.3i)19-s − 12.1i·20-s − 17.1·22-s + (−10.5 − 6.06i)23-s + (5.91 + 10.2i)25-s + (22.7 + 13.1i)26-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−1.05 − 0.606i)5-s − 0.353i·8-s + (0.429 + 0.743i)10-s + (0.955 − 0.551i)11-s − 1.42·13-s + (−0.125 + 0.216i)16-s + (0.555 − 0.320i)17-s + (−0.526 + 0.911i)19-s − 0.606i·20-s − 0.780·22-s + (−0.457 − 0.263i)23-s + (0.236 + 0.409i)25-s + (0.875 + 0.505i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3681097541\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3681097541\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.22 + 0.707i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (5.25 + 3.03i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-10.5 + 6.06i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + (-9.44 + 5.45i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (10 - 17.3i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (10.5 + 6.06i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 41.8iT - 841T^{2} \) |
| 31 | \( 1 + (12.5 + 21.7i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (19 - 32.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 60.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 83.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (14.6 + 8.48i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (81.4 - 47.0i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-50.4 + 29.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (7.83 - 13.5i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-66.3 - 114. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 12.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-38.4 - 66.6i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (16.8 - 29.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 60.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (4.13 + 2.38i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 188.T + 9.40e3T^{2} \) |
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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
−9.864992327901524966868235853001, −9.450462867113155320721190274241, −8.205114330771295709446324907720, −7.989619909085208810306753251076, −6.93006348284292888177398931479, −5.82825660005679381981411074619, −4.47631855129915451900886655202, −3.82273639246621460262055251201, −2.51136934897904747840970781117, −1.00578862342427082412136940712,
0.17978572353605777820556918976, 1.92855990831688585391888783216, 3.29408018365350200546128548804, 4.34838473573711241178627377968, 5.41228859679543252373180768191, 6.78773370949926114136669377360, 7.17374714666712562128225562142, 7.901363415866070024076894295133, 8.981616921572864649106338210682, 9.597845333395858164695457752129