| L(s) = 1 | + (1.98 + 0.174i)2-s + (−1.25 − 1.19i)3-s + (1.95 + 0.344i)4-s + (0.840 − 2.07i)5-s + (−2.28 − 2.59i)6-s + (1.15 + 0.808i)7-s + (−0.0258 − 0.00692i)8-s + (0.145 + 2.99i)9-s + (2.03 − 3.97i)10-s + (−1.21 + 3.35i)11-s + (−2.04 − 2.76i)12-s + (0.265 + 3.03i)13-s + (2.15 + 1.80i)14-s + (−3.52 + 1.59i)15-s + (−3.78 − 1.37i)16-s + (3.91 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (1.40 + 0.123i)2-s + (−0.724 − 0.689i)3-s + (0.978 + 0.172i)4-s + (0.376 − 0.926i)5-s + (−0.933 − 1.05i)6-s + (0.436 + 0.305i)7-s + (−0.00914 − 0.00244i)8-s + (0.0484 + 0.998i)9-s + (0.642 − 1.25i)10-s + (−0.367 + 1.01i)11-s + (−0.589 − 0.799i)12-s + (0.0737 + 0.842i)13-s + (0.576 + 0.483i)14-s + (−0.911 + 0.411i)15-s + (−0.945 − 0.344i)16-s + (0.950 − 0.254i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.854 + 0.519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69886 - 0.476078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69886 - 0.476078i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.25 + 1.19i)T \) |
| 5 | \( 1 + (-0.840 + 2.07i)T \) |
| good | 2 | \( 1 + (-1.98 - 0.174i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 0.808i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (1.21 - 3.35i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.265 - 3.03i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-3.91 + 1.05i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.42 - 0.821i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 + 5.24i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (2.12 - 1.78i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.13 - 6.43i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.751 - 2.80i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.92 + 7.06i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.41 - 5.17i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-4.56 + 6.52i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-8.37 + 8.37i)T - 53iT^{2} \) |
| 59 | \( 1 + (13.4 - 4.89i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 6.32i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (11.6 - 1.01i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.966 - 0.557i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.19 + 8.19i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.51 + 4.18i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.529 + 6.05i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (0.0742 + 0.128i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.6 + 5.45i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.01971255721253530907636168045, −12.24765328901867144901258068616, −11.88392696240938791564638303440, −10.32240484711264922427189991643, −8.829733126953241338944653106028, −7.38712536238043171275751137871, −6.15352467556649027243961792906, −5.21379409339657906329108433876, −4.42944365177569789968017185609, −2.04712134052848917629674255759,
3.05938548986847178997166607985, 4.11455325961876763027959078970, 5.68414694011130067884274666831, 5.95145812848505878385574258359, 7.68848272674137241493775520390, 9.543709507040703586540211541302, 10.77834131075906529443817012864, 11.21681605081169188131842262632, 12.34572700901113212290064348055, 13.46307459024395912446850386626
