| L(s) = 1 | + (−1.52 − 2.18i)2-s + (1.25 + 1.19i)3-s + (−1.73 + 4.77i)4-s + (−1.77 + 1.36i)5-s + (0.674 − 4.56i)6-s + (2.78 + 1.30i)7-s + (7.92 − 2.12i)8-s + (0.166 + 2.99i)9-s + (5.67 + 1.78i)10-s + (−0.426 + 0.508i)11-s + (−7.87 + 3.94i)12-s + (0.384 + 0.269i)13-s + (−1.42 − 8.06i)14-s + (−3.85 − 0.398i)15-s + (−8.94 − 7.50i)16-s + (4.50 + 1.20i)17-s + ⋯ |
| L(s) = 1 | + (−1.07 − 1.54i)2-s + (0.726 + 0.687i)3-s + (−0.869 + 2.38i)4-s + (−0.793 + 0.608i)5-s + (0.275 − 1.86i)6-s + (1.05 + 0.491i)7-s + (2.80 − 0.750i)8-s + (0.0555 + 0.998i)9-s + (1.79 + 0.565i)10-s + (−0.128 + 0.153i)11-s + (−2.27 + 1.13i)12-s + (0.106 + 0.0746i)13-s + (−0.380 − 2.15i)14-s + (−0.994 − 0.102i)15-s + (−2.23 − 1.87i)16-s + (1.09 + 0.292i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.732191 - 0.0277023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.732191 - 0.0277023i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.77 - 1.36i)T \) |
| good | 2 | \( 1 + (1.52 + 2.18i)T + (-0.684 + 1.87i)T^{2} \) |
| 7 | \( 1 + (-2.78 - 1.30i)T + (4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (0.426 - 0.508i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.384 - 0.269i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-4.50 - 1.20i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (4.80 + 2.77i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.0280 - 0.0602i)T + (-14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.434 - 2.46i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.76 - 0.642i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-0.656 + 2.44i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-1.62 + 0.286i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-8.36 + 0.732i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-1.73 + 3.71i)T + (-30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (7.52 + 7.52i)T + 53iT^{2} \) |
| 59 | \( 1 + (-3.49 + 2.93i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.84 - 2.12i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-3.13 + 4.48i)T + (-22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (5.33 - 3.07i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.64 + 13.6i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.34 - 0.941i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.64 + 3.25i)T + (28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-3.08 + 5.33i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.18 + 13.5i)T + (-95.5 + 16.8i)T^{2} \) |
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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
−12.80306376581485945406826271854, −11.78810293389628793229524765726, −10.93832348886930851714143803319, −10.36193175289708762265142243890, −9.123362382005509469577437913201, −8.297610270088843512423500424587, −7.60346387548222512192842939636, −4.58072663327386020507672266892, −3.41923813845041007426958547411, −2.18285642919872817445998970347,
1.14754873351708763929688119843, 4.40069658216692082671206072766, 5.89968131611204992419675879255, 7.31571498541181897243427521087, 7.959404345363041934202024449344, 8.469918029902587600894519654034, 9.588657784854065647126596061969, 10.93801591811840753376684777694, 12.38391044290875324691069048664, 13.74388987926351911888576871099
