| L(s) = 1 | + (−2.49 + 0.440i)2-s + (2.49 + 1.66i)3-s + (2.28 − 0.831i)4-s + (−5.84 − 6.96i)5-s + (−6.96 − 3.06i)6-s + (1.19 + 2.07i)7-s + (3.44 − 1.98i)8-s + (3.45 + 8.31i)9-s + (17.6 + 14.8i)10-s + 19.3i·11-s + (7.08 + 1.73i)12-s + (9.11 + 7.64i)13-s + (−3.91 − 4.65i)14-s + (−2.98 − 27.0i)15-s + (−15.1 + 12.7i)16-s + (−8.83 − 10.5i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 + 0.220i)2-s + (0.831 + 0.555i)3-s + (0.571 − 0.207i)4-s + (−1.16 − 1.39i)5-s + (−1.16 − 0.510i)6-s + (0.171 + 0.296i)7-s + (0.430 − 0.248i)8-s + (0.383 + 0.923i)9-s + (1.76 + 1.48i)10-s + 1.75i·11-s + (0.590 + 0.144i)12-s + (0.701 + 0.588i)13-s + (−0.279 − 0.332i)14-s + (−0.198 − 1.80i)15-s + (−0.948 + 0.795i)16-s + (−0.519 − 0.619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.430440 + 0.536008i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.430440 + 0.536008i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.49 - 1.66i)T \) |
| 19 | \( 1 + (3.04 - 18.7i)T \) |
| good | 2 | \( 1 + (2.49 - 0.440i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (5.84 + 6.96i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-1.19 - 2.07i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 - 19.3iT - 121T^{2} \) |
| 13 | \( 1 + (-9.11 - 7.64i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (8.83 + 10.5i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (-2.19 - 6.04i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (-1.07 - 2.95i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 - 34.3T + 961T^{2} \) |
| 37 | \( 1 + 42.9T + 1.36e3T^{2} \) |
| 41 | \( 1 + (29.8 - 5.25i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-42.6 - 15.5i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-19.2 - 53.0i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-53.5 - 9.44i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (20.5 - 56.3i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (63.6 + 53.4i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-0.384 + 2.18i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (25.8 - 4.56i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (11.2 + 4.07i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-78.8 + 66.1i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (25.4 - 14.6i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (10.0 + 27.6i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (6.31 + 35.7i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.67919380722597314228928182424, −11.76748073256421725590006164504, −10.39490872486655805776334238075, −9.349329310467655193297353034705, −8.818123492748181193364909590420, −8.002375920730684440717017614102, −7.23436965649824005449030512914, −4.75424757050149686197423997454, −4.10414343674793530430987868562, −1.62657796596496604487221262547,
0.61183336802263597786502854700, 2.73542869887991018961996414395, 3.81746974306520969686702032082, 6.45777001195553396903815439007, 7.40648381190921219875954991916, 8.326604967373747700210142845660, 8.751105946086159930937150360270, 10.47250882015755601233850441708, 10.90411720141734762799719800313, 11.82502993424846296865286455690
