| L(s) = 1 | + (−0.639 − 0.0559i)2-s + (1.67 + 0.432i)3-s + (−1.56 − 0.275i)4-s + (−1.26 + 1.84i)5-s + (−1.04 − 0.370i)6-s + (3.52 + 2.47i)7-s + (2.22 + 0.595i)8-s + (2.62 + 1.45i)9-s + (0.910 − 1.10i)10-s + (−0.157 + 0.433i)11-s + (−2.50 − 1.13i)12-s + (−0.462 − 5.28i)13-s + (−2.11 − 1.77i)14-s + (−2.91 + 2.54i)15-s + (1.59 + 0.580i)16-s + (−4.93 + 1.32i)17-s + ⋯ |
| L(s) = 1 | + (−0.452 − 0.0395i)2-s + (0.968 + 0.249i)3-s + (−0.782 − 0.137i)4-s + (−0.564 + 0.825i)5-s + (−0.427 − 0.151i)6-s + (1.33 + 0.933i)7-s + (0.786 + 0.210i)8-s + (0.875 + 0.483i)9-s + (0.287 − 0.350i)10-s + (−0.0476 + 0.130i)11-s + (−0.722 − 0.328i)12-s + (−0.128 − 1.46i)13-s + (−0.565 − 0.474i)14-s + (−0.752 + 0.658i)15-s + (0.399 + 0.145i)16-s + (−1.19 + 0.320i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.721 - 0.692i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.946680 + 0.381131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.946680 + 0.381131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.67 - 0.432i)T \) |
| 5 | \( 1 + (1.26 - 1.84i)T \) |
| good | 2 | \( 1 + (0.639 + 0.0559i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-3.52 - 2.47i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (0.157 - 0.433i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.462 + 5.28i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (4.93 - 1.32i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 0.598i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.57 + 2.25i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (0.00462 - 0.00387i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.387 - 2.19i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.94 + 7.25i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-3.02 + 3.60i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.806 - 1.72i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-6.47 + 9.24i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (-3.94 + 3.94i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.388 - 0.141i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.399 - 2.26i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (5.84 - 0.511i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.648 - 0.374i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.97 - 11.1i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.01 + 1.20i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.930 + 10.6i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-6.36 - 11.0i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.76 - 3.62i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.57377369756060313509073648438, −12.40369822218108385144321064010, −10.96155923183118975626448906038, −10.28267145811290794092440604018, −8.891467934391980424387292438688, −8.287440703343834310395363472928, −7.44494558305422744279845091403, −5.30482295493619505868483234997, −4.06935336168733940411359912617, −2.36810852513663866258867429737,
1.45088729275704099484591052796, 4.13339481670995486531508307765, 4.61387734485781726758584748056, 7.21986090446691250046904319805, 7.983419220552291592220046360647, 8.797804508782882646906924388196, 9.563472730540142099330253456975, 11.05765151629239571353706021017, 12.16017236297285357076288985750, 13.55676393327164560037381417091
