| L(s) = 1 | + (−0.156 + 0.987i)2-s + (2.10 + 1.07i)3-s + (−0.951 − 0.309i)4-s + (−0.507 + 2.17i)5-s + (−1.38 + 1.91i)6-s + (−1.71 − 3.37i)7-s + (0.453 − 0.891i)8-s + (1.52 + 2.09i)9-s + (−2.07 − 0.842i)10-s + (2.91 − 1.58i)11-s + (−1.67 − 1.67i)12-s + (−2.38 − 0.378i)13-s + (3.60 − 1.16i)14-s + (−3.40 + 4.04i)15-s + (0.809 + 0.587i)16-s + (1.89 − 0.299i)17-s + ⋯ |
| L(s) = 1 | + (−0.110 + 0.698i)2-s + (1.21 + 0.619i)3-s + (−0.475 − 0.154i)4-s + (−0.227 + 0.973i)5-s + (−0.567 + 0.780i)6-s + (−0.649 − 1.27i)7-s + (0.160 − 0.315i)8-s + (0.507 + 0.698i)9-s + (−0.655 − 0.266i)10-s + (0.877 − 0.479i)11-s + (−0.482 − 0.482i)12-s + (−0.662 − 0.104i)13-s + (0.962 − 0.312i)14-s + (−0.879 + 1.04i)15-s + (0.202 + 0.146i)16-s + (0.458 − 0.0726i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.949588 + 0.771876i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.949588 + 0.771876i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.156 - 0.987i)T \) |
| 5 | \( 1 + (0.507 - 2.17i)T \) |
| 11 | \( 1 + (-2.91 + 1.58i)T \) |
| good | 3 | \( 1 + (-2.10 - 1.07i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (1.71 + 3.37i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (2.38 + 0.378i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.89 + 0.299i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.20 - 3.70i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.12 - 2.12i)T - 23iT^{2} \) |
| 29 | \( 1 + (-2.45 + 7.56i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.04 + 2.21i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (5.50 - 2.80i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (10.5 - 3.44i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (0.0971 + 0.0971i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.20 - 8.25i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.201 + 1.27i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-13.8 - 4.50i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.01 - 2.77i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (7.14 + 7.14i)T + 67iT^{2} \) |
| 71 | \( 1 + (-10.0 - 7.32i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.92 - 4.54i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (3.21 - 2.33i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-0.0974 - 0.615i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 2.93iT - 89T^{2} \) |
| 97 | \( 1 + (-5.70 - 0.904i)T + (92.2 + 29.9i)T^{2} \) |
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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
−14.07575213645736083987531274704, −13.56866953510021783486932897435, −11.74401717220755850677576635683, −9.986500197109673275221796298333, −9.934495275901681517801472916609, −8.309358281554144811731948348939, −7.40483957053081027032516056665, −6.31461101260356960812826514458, −4.09709517080564801520973332126, −3.27091094265569447327012688106,
1.95016471854904808292309353863, 3.32650012813944895241567392732, 5.08402128060083438869458323226, 7.00635472381967175618031810563, 8.524469344143284911850242101111, 8.929873559908764396853609832952, 9.877625306786172866281134714500, 11.94071695171309323576655381466, 12.33120067351111925420582043610, 13.22156560194651805516190415214
