| L(s) = 1 | + (−1.39 + 0.221i)2-s + (0.786 − 1.54i)3-s + (1.90 − 0.618i)4-s + (2.03 + 4.56i)5-s + (−0.756 + 2.32i)6-s + (−7.48 + 7.48i)7-s + (−2.52 + 1.28i)8-s + (−1.76 − 2.42i)9-s + (−3.84 − 5.93i)10-s + (6.98 + 5.07i)11-s + (0.541 − 3.42i)12-s + (−2.37 − 0.376i)13-s + (8.79 − 12.1i)14-s + (8.64 + 0.456i)15-s + (3.23 − 2.35i)16-s + (14.2 + 27.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.698 + 0.110i)2-s + (0.262 − 0.514i)3-s + (0.475 − 0.154i)4-s + (0.406 + 0.913i)5-s + (−0.126 + 0.388i)6-s + (−1.06 + 1.06i)7-s + (−0.315 + 0.160i)8-s + (−0.195 − 0.269i)9-s + (−0.384 − 0.593i)10-s + (0.635 + 0.461i)11-s + (0.0451 − 0.285i)12-s + (−0.182 − 0.0289i)13-s + (0.628 − 0.865i)14-s + (0.576 + 0.0304i)15-s + (0.202 − 0.146i)16-s + (0.838 + 1.64i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.788206 + 0.624191i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.788206 + 0.624191i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 - 0.221i)T \) |
| 3 | \( 1 + (-0.786 + 1.54i)T \) |
| 5 | \( 1 + (-2.03 - 4.56i)T \) |
| good | 7 | \( 1 + (7.48 - 7.48i)T - 49iT^{2} \) |
| 11 | \( 1 + (-6.98 - 5.07i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (2.37 + 0.376i)T + (160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-14.2 - 27.9i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (6.61 + 2.15i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-1.07 - 6.79i)T + (-503. + 163. i)T^{2} \) |
| 29 | \( 1 + (-27.9 + 9.07i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (4.28 - 13.1i)T + (-777. - 564. i)T^{2} \) |
| 37 | \( 1 + (-3.51 + 22.2i)T + (-1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-50.3 + 36.5i)T + (519. - 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-6.29 - 6.29i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (33.0 + 16.8i)T + (1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (27.5 - 54.1i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (66.1 + 91.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-15.8 - 11.5i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (57.4 + 112. i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-27.9 - 85.9i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (12.1 + 77.0i)T + (-5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (-89.1 + 28.9i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (36.9 - 18.8i)T + (4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (51.2 - 70.5i)T + (-2.44e3 - 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-124. - 63.1i)T + (5.53e3 + 7.61e3i)T^{2} \) |
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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
−12.73719751899717071454902708611, −12.14740815853823700582189374602, −10.76145726758793574746411241098, −9.772194227809392587830072880381, −8.985783022783554467878630996775, −7.73295732603679582372755600571, −6.51422693705864357648030283726, −5.96986764805699973961082761499, −3.31427456920120281203063364409, −2.02524332541240198793830581123,
0.800145315781565046913332380287, 3.09741144081037376782544895972, 4.54119513757573528573970512415, 6.16298175581565062429540349055, 7.41313908971484748740465804153, 8.688740796247908152787353573246, 9.630407263236416865404107730184, 10.08074699675547009423148036775, 11.41567055645274108348797922940, 12.55088969936759819841787227831
