| L(s) = 1 | + (−0.641 + 1.11i)2-s + (−2.04 + 0.745i)3-s + (0.178 + 0.308i)4-s + (−2.94 − 2.46i)5-s + (0.485 − 2.75i)6-s + (0.512 + 0.430i)7-s − 3.02·8-s + (1.34 − 1.12i)9-s + (4.62 − 1.68i)10-s + (0.102 + 0.0374i)11-s + (−0.594 − 0.499i)12-s + (−2.65 − 2.22i)13-s + (−0.806 + 0.293i)14-s + (7.87 + 2.86i)15-s + (1.58 − 2.73i)16-s + (−3.00 + 5.19i)17-s + ⋯ |
| L(s) = 1 | + (−0.453 + 0.785i)2-s + (−1.18 + 0.430i)3-s + (0.0890 + 0.154i)4-s + (−1.31 − 1.10i)5-s + (0.198 − 1.12i)6-s + (0.193 + 0.162i)7-s − 1.06·8-s + (0.447 − 0.375i)9-s + (1.46 − 0.532i)10-s + (0.0310 + 0.0112i)11-s + (−0.171 − 0.144i)12-s + (−0.735 − 0.617i)13-s + (−0.215 + 0.0784i)14-s + (2.03 + 0.739i)15-s + (0.395 − 0.684i)16-s + (−0.728 + 1.26i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0422781 - 0.122849i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0422781 - 0.122849i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 109 | \( 1 + (-4.16 + 9.57i)T \) |
| good | 2 | \( 1 + (0.641 - 1.11i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (2.04 - 0.745i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (2.94 + 2.46i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.512 - 0.430i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.102 - 0.0374i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.65 + 2.22i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (3.00 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.68 - 6.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.750 - 1.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.770 + 0.280i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.08 + 5.94i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (5.99 - 5.03i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + 1.43T + 41T^{2} \) |
| 43 | \( 1 + (3.35 - 5.81i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.932 + 5.28i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.582 - 0.489i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (1.12 + 0.407i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.403 + 2.28i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.95 + 4.99i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.81 - 11.8i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.06 + 2.93i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (8.28 - 6.94i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.52 + 1.28i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (1.50 - 8.52i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.93 + 2.46i)T + (16.8 + 95.5i)T^{2} \) |
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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
−15.05043522984739187086472017613, −12.79751998975923898627603991780, −12.10684457772501867229835801906, −11.42823829693132276525792341719, −10.09655288030160869417602781593, −8.415955288059138567586564791351, −8.047833263608275801970649539641, −6.47082468574488129720910443114, −5.26220839182201743616063545556, −4.04054198899157291279588165264,
0.18347516943301467552413591286, 2.76311317856765019666935224409, 4.74193933498280766456291629878, 6.63317556304266180517619898621, 7.08204908853496461581044400837, 8.889945113592551169464512376588, 10.44517572008020955527402800845, 11.15570185931273399394876611347, 11.65446397030825164557998556898, 12.37476749946156426292713069921
