| L(s) = 1 | + (−1.00 + 2.99i)2-s + (−2.99 − 0.116i)3-s + (−4.74 − 3.60i)4-s + (−2.80 − 1.11i)5-s + (3.36 − 8.84i)6-s + (−12.2 + 5.68i)7-s + (5.12 − 3.47i)8-s + (8.97 + 0.698i)9-s + (6.17 − 7.26i)10-s + (15.0 + 4.18i)11-s + (13.8 + 11.3i)12-s + (9.63 − 18.1i)13-s + (−4.62 − 42.5i)14-s + (8.28 + 3.67i)15-s + (−1.14 − 4.14i)16-s + (−5.45 + 11.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.503 + 1.49i)2-s + (−0.999 − 0.0388i)3-s + (−1.18 − 0.902i)4-s + (−0.561 − 0.223i)5-s + (0.561 − 1.47i)6-s + (−1.75 + 0.812i)7-s + (0.640 − 0.434i)8-s + (0.996 + 0.0775i)9-s + (0.617 − 0.726i)10-s + (1.37 + 0.380i)11-s + (1.15 + 0.947i)12-s + (0.740 − 1.39i)13-s + (−0.330 − 3.03i)14-s + (0.552 + 0.245i)15-s + (−0.0718 − 0.258i)16-s + (−0.320 + 0.692i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0248i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.302337 - 0.00375980i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.302337 - 0.00375980i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.99 + 0.116i)T \) |
| 59 | \( 1 + (-42.7 + 40.6i)T \) |
| good | 2 | \( 1 + (1.00 - 2.99i)T + (-3.18 - 2.42i)T^{2} \) |
| 5 | \( 1 + (2.80 + 1.11i)T + (18.1 + 17.1i)T^{2} \) |
| 7 | \( 1 + (12.2 - 5.68i)T + (31.7 - 37.3i)T^{2} \) |
| 11 | \( 1 + (-15.0 - 4.18i)T + (103. + 62.3i)T^{2} \) |
| 13 | \( 1 + (-9.63 + 18.1i)T + (-94.8 - 139. i)T^{2} \) |
| 17 | \( 1 + (5.45 - 11.7i)T + (-187. - 220. i)T^{2} \) |
| 19 | \( 1 + (-8.96 + 1.97i)T + (327. - 151. i)T^{2} \) |
| 23 | \( 1 + (0.699 + 0.114i)T + (501. + 168. i)T^{2} \) |
| 29 | \( 1 + (5.03 + 14.9i)T + (-669. + 508. i)T^{2} \) |
| 31 | \( 1 + (14.1 + 3.12i)T + (872. + 403. i)T^{2} \) |
| 37 | \( 1 + (-20.7 + 30.5i)T + (-506. - 1.27e3i)T^{2} \) |
| 41 | \( 1 + (49.6 - 8.13i)T + (1.59e3 - 536. i)T^{2} \) |
| 43 | \( 1 + (4.03 + 14.5i)T + (-1.58e3 + 953. i)T^{2} \) |
| 47 | \( 1 + (28.7 - 11.4i)T + (1.60e3 - 1.51e3i)T^{2} \) |
| 53 | \( 1 + (-54.4 + 46.2i)T + (454. - 2.77e3i)T^{2} \) |
| 61 | \( 1 + (59.1 + 19.9i)T + (2.96e3 + 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-41.2 - 60.8i)T + (-1.66e3 + 4.17e3i)T^{2} \) |
| 71 | \( 1 + (-57.6 + 22.9i)T + (3.65e3 - 3.46e3i)T^{2} \) |
| 73 | \( 1 + (40.7 - 4.43i)T + (5.20e3 - 1.14e3i)T^{2} \) |
| 79 | \( 1 + (-77.8 + 46.8i)T + (2.92e3 - 5.51e3i)T^{2} \) |
| 83 | \( 1 + (112. - 6.07i)T + (6.84e3 - 744. i)T^{2} \) |
| 89 | \( 1 + (-4.43 - 13.1i)T + (-6.30e3 + 4.79e3i)T^{2} \) |
| 97 | \( 1 + (-8.85 - 0.962i)T + (9.18e3 + 2.02e3i)T^{2} \) |
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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
−12.49670770850888591739694644750, −11.62921326850147492721204277909, −10.07824251126910027629402381757, −9.295276888751614447717946958639, −8.227213141951255431216102826222, −6.93434005372955667565716011462, −6.23578556995501443833516131628, −5.53632788748770602037490018821, −3.79005525884316273099817915920, −0.29341600868681229567660528002,
1.15932475105979422572815016010, 3.48885539273128861180635135456, 4.07890002746820415498974619724, 6.36828400772336581943451047701, 7.00085240631617776477502205939, 9.108744540801356208712345116850, 9.652282377073783722492202280160, 10.65448491124266901775162674582, 11.57576987651449736239691088300, 11.91004442681865471847899817446
