| L(s) = 1 | + (−3.88 − 0.772i)3-s + (2.37 + 3.55i)5-s + (2.98 − 4.47i)7-s + (6.17 + 2.55i)9-s + (−4.17 − 21.0i)11-s + (−13.3 − 13.3i)13-s + (−6.47 − 15.6i)15-s + (16.7 − 2.69i)17-s + (3.71 − 1.53i)19-s + (−15.0 + 15.0i)21-s + (−8.85 + 1.76i)23-s + (2.57 − 6.22i)25-s + (7.62 + 5.09i)27-s + (13.2 − 8.82i)29-s + (−5.26 + 26.4i)31-s + ⋯ |
| L(s) = 1 | + (−1.29 − 0.257i)3-s + (0.474 + 0.710i)5-s + (0.427 − 0.639i)7-s + (0.686 + 0.284i)9-s + (−0.379 − 1.91i)11-s + (−1.02 − 1.02i)13-s + (−0.431 − 1.04i)15-s + (0.987 − 0.158i)17-s + (0.195 − 0.0808i)19-s + (−0.717 + 0.717i)21-s + (−0.384 + 0.0765i)23-s + (0.103 − 0.248i)25-s + (0.282 + 0.188i)27-s + (0.455 − 0.304i)29-s + (−0.169 + 0.854i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0471 + 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0471 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.556444 - 0.583325i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556444 - 0.583325i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + (-16.7 + 2.69i)T \) |
| good | 3 | \( 1 + (3.88 + 0.772i)T + (8.31 + 3.44i)T^{2} \) |
| 5 | \( 1 + (-2.37 - 3.55i)T + (-9.56 + 23.0i)T^{2} \) |
| 7 | \( 1 + (-2.98 + 4.47i)T + (-18.7 - 45.2i)T^{2} \) |
| 11 | \( 1 + (4.17 + 21.0i)T + (-111. + 46.3i)T^{2} \) |
| 13 | \( 1 + (13.3 + 13.3i)T + 169iT^{2} \) |
| 19 | \( 1 + (-3.71 + 1.53i)T + (255. - 255. i)T^{2} \) |
| 23 | \( 1 + (8.85 - 1.76i)T + (488. - 202. i)T^{2} \) |
| 29 | \( 1 + (-13.2 + 8.82i)T + (321. - 776. i)T^{2} \) |
| 31 | \( 1 + (5.26 - 26.4i)T + (-887. - 367. i)T^{2} \) |
| 37 | \( 1 + (9.96 + 1.98i)T + (1.26e3 + 523. i)T^{2} \) |
| 41 | \( 1 + (-27.7 + 41.5i)T + (-643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (59.9 + 24.8i)T + (1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (26.7 + 26.7i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (51.4 - 21.3i)T + (1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (12.3 - 29.8i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-30.4 - 20.3i)T + (1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 - 61.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-105. - 21.0i)T + (4.65e3 + 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-13.4 - 20.0i)T + (-2.03e3 + 4.92e3i)T^{2} \) |
| 79 | \( 1 + (-19.4 - 97.8i)T + (-5.76e3 + 2.38e3i)T^{2} \) |
| 83 | \( 1 + (43.3 + 104. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-72.1 + 72.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (75.6 - 50.5i)T + (3.60e3 - 8.69e3i)T^{2} \) |
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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
−12.54847356085493210777934625627, −11.56609872341865627632975231424, −10.67739515939512680877643932353, −10.16560663049867899592361738220, −8.263678592720154503125748326858, −7.11103579114438608585235753629, −5.93139548569373754515559431399, −5.21606710515684770296630039625, −3.11462264725809416811960194598, −0.63779235067807930912800447770,
1.88756708102775819920020478653, 4.76247583831211932697223332749, 5.12805110842361285462947103406, 6.49220613055364765864322210851, 7.82469861201490340262881329092, 9.484009392966062155617957067372, 10.00418594944017916156022540247, 11.43515707190217925141051100753, 12.21257199240611165955573436508, 12.74912017018907626433369020371
