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
−12.37476749946156426292713069921, −11.65446397030825164557998556898, −11.15570185931273399394876611347, −10.44517572008020955527402800845, −8.889945113592551169464512376588, −7.08204908853496461581044400837, −6.63317556304266180517619898621, −4.74193933498280766456291629878, −2.76311317856765019666935224409, −0.18347516943301467552413591286,
4.04054198899157291279588165264, 5.26220839182201743616063545556, 6.47082468574488129720910443114, 8.047833263608275801970649539641, 8.415955288059138567586564791351, 10.09655288030160869417602781593, 11.42823829693132276525792341719, 12.10684457772501867229835801906, 12.79751998975923898627603991780, 15.05043522984739187086472017613