L(s) = 1 | + (0.884 + 1.10i)2-s + (−1.18 + 1.26i)3-s + (−0.436 + 1.95i)4-s + (0.5 − 0.866i)5-s + (−2.44 − 0.193i)6-s + (−2.20 + 1.27i)7-s + (−2.54 + 1.24i)8-s + (−0.185 − 2.99i)9-s + (1.39 − 0.213i)10-s + (−4.02 + 2.32i)11-s + (−1.94 − 2.86i)12-s + (3.23 + 1.87i)13-s + (−3.35 − 1.30i)14-s + (0.499 + 1.65i)15-s + (−3.61 − 1.70i)16-s + 3.38i·17-s + ⋯ |
L(s) = 1 | + (0.625 + 0.780i)2-s + (−0.684 + 0.728i)3-s + (−0.218 + 0.975i)4-s + (0.223 − 0.387i)5-s + (−0.996 − 0.0790i)6-s + (−0.834 + 0.481i)7-s + (−0.898 + 0.439i)8-s + (−0.0616 − 0.998i)9-s + (0.442 − 0.0676i)10-s + (−1.21 + 0.701i)11-s + (−0.561 − 0.827i)12-s + (0.898 + 0.518i)13-s + (−0.897 − 0.350i)14-s + (0.129 + 0.428i)15-s + (−0.904 − 0.425i)16-s + 0.821i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0819104 - 0.972042i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0819104 - 0.972042i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.884 - 1.10i)T \) |
| 3 | \( 1 + (1.18 - 1.26i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
good | 7 | \( 1 + (2.20 - 1.27i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.02 - 2.32i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.23 - 1.87i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 3.38iT - 17T^{2} \) |
| 19 | \( 1 + 5.12T + 19T^{2} \) |
| 23 | \( 1 + (-4.16 + 7.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.01 - 3.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.88 - 2.81i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.40iT - 37T^{2} \) |
| 41 | \( 1 + (5.23 + 3.02i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.26 - 9.11i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.621 - 1.07i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.05T + 53T^{2} \) |
| 59 | \( 1 + (0.351 + 0.202i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.57 - 2.64i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.17 - 10.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.43T + 71T^{2} \) |
| 73 | \( 1 - 13.3T + 73T^{2} \) |
| 79 | \( 1 + (-5.40 + 3.11i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-12.0 + 6.97i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.72iT - 89T^{2} \) |
| 97 | \( 1 + (-1.31 - 2.27i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.37307225629358957282894944322, −10.96076579666978098904354406140, −10.14454588155614519248632458024, −8.980195502594808151509591521441, −8.329719070396318947860751052521, −6.63606612295618832184302566923, −6.19962654620009329415832868251, −5.03521994697749524248614899342, −4.29901101943148576032549579370, −2.89039164528176000762507190572,
0.57385558354251771192100431296, 2.43258799816357287057320704301, 3.55355037822099307445296895504, 5.16024422104248633471356744002, 5.97201303197484429576175633615, 6.79595875472793271825732076105, 8.016955348555688499577847887740, 9.440796757066990159118089150541, 10.67605043916397973207415979873, 10.77583286334407688364163473335