L(s) = 1 | + (1.48 + 2.57i)2-s + (−1.79 + 2.40i)3-s + (−2.43 + 4.22i)4-s + (−3.23 + 3.80i)5-s + (−8.87 − 1.06i)6-s + (5.16 − 4.72i)7-s − 2.60·8-s + (−2.53 − 8.63i)9-s + (−14.6 − 2.68i)10-s + (−3.56 − 2.05i)11-s + (−5.75 − 13.4i)12-s + 21.3i·13-s + (19.8 + 6.30i)14-s + (−3.32 − 14.6i)15-s + (5.87 + 10.1i)16-s + (1.11 − 1.92i)17-s + ⋯ |
L(s) = 1 | + (0.744 + 1.28i)2-s + (−0.599 + 0.800i)3-s + (−0.609 + 1.05i)4-s + (−0.647 + 0.761i)5-s + (−1.47 − 0.176i)6-s + (0.738 − 0.674i)7-s − 0.325·8-s + (−0.281 − 0.959i)9-s + (−1.46 − 0.268i)10-s + (−0.323 − 0.186i)11-s + (−0.479 − 1.11i)12-s + 1.64i·13-s + (1.41 + 0.450i)14-s + (−0.221 − 0.975i)15-s + (0.367 + 0.635i)16-s + (0.0653 − 0.113i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.989 - 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.109790 + 1.48960i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.109790 + 1.48960i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.79 - 2.40i)T \) |
| 5 | \( 1 + (3.23 - 3.80i)T \) |
| 7 | \( 1 + (-5.16 + 4.72i)T \) |
good | 2 | \( 1 + (-1.48 - 2.57i)T + (-2 + 3.46i)T^{2} \) |
| 11 | \( 1 + (3.56 + 2.05i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 21.3iT - 169T^{2} \) |
| 17 | \( 1 + (-1.11 + 1.92i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-4.93 - 8.55i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-9.66 - 16.7i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 28.1iT - 841T^{2} \) |
| 31 | \( 1 + (-17.6 + 30.5i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-43.1 + 24.9i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 27.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 41.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (19.1 + 33.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-25.6 + 44.4i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-22.0 - 12.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (34.1 + 59.2i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-59.1 - 34.1i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 11.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (88.6 + 51.1i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-32.1 - 55.7i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 122.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-116. + 67.2i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 13.6iT - 9.40e3T^{2} \) |
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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
−14.43417205947112842229622651175, −13.45822235443613584012598511926, −11.69010874451277854180289454170, −11.12774368309150977021652462925, −9.807338245713085225655631679219, −8.092993992637788550907315660998, −7.08569245479341663557123681823, −6.06847833375334732573277113319, −4.66043885518469251328931631185, −3.89582835566798559926186319547,
1.07440162696704758806588195137, 2.79490619265118629517914206702, 4.76349425342028937981427854928, 5.47266329689364508350105926224, 7.55529609885061713888156305304, 8.559513105344879870623651014746, 10.45268037358876074268260773379, 11.24886786923054780358080296291, 12.22143803587962925307108151871, 12.62188429176543034758636366080