| L(s) = 1 | + (0.156 − 0.987i)2-s + (2.86 + 1.45i)3-s + (−0.951 − 0.309i)4-s + (−1.90 − 1.17i)5-s + (1.88 − 2.59i)6-s + (−0.692 − 1.35i)7-s + (−0.453 + 0.891i)8-s + (4.29 + 5.91i)9-s + (−1.45 + 1.69i)10-s + (−2.70 + 1.91i)11-s + (−2.27 − 2.27i)12-s + (1.04 + 0.165i)13-s + (−1.44 + 0.471i)14-s + (−3.72 − 6.13i)15-s + (0.809 + 0.587i)16-s + (−2.19 + 0.347i)17-s + ⋯ |
| L(s) = 1 | + (0.110 − 0.698i)2-s + (1.65 + 0.841i)3-s + (−0.475 − 0.154i)4-s + (−0.850 − 0.526i)5-s + (0.770 − 1.06i)6-s + (−0.261 − 0.513i)7-s + (−0.160 + 0.315i)8-s + (1.43 + 1.97i)9-s + (−0.461 + 0.535i)10-s + (−0.815 + 0.578i)11-s + (−0.655 − 0.655i)12-s + (0.289 + 0.0459i)13-s + (−0.387 + 0.125i)14-s + (−0.961 − 1.58i)15-s + (0.202 + 0.146i)16-s + (−0.532 + 0.0843i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37705 - 0.288738i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37705 - 0.288738i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.90 + 1.17i)T \) |
| 11 | \( 1 + (2.70 - 1.91i)T \) |
| good | 3 | \( 1 + (-2.86 - 1.45i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (0.692 + 1.35i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (-1.04 - 0.165i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (2.19 - 0.347i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.91 + 5.89i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (3.05 - 3.05i)T - 23iT^{2} \) |
| 29 | \( 1 + (-1.39 + 4.30i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 1.69i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.24 + 0.631i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.38 + 0.774i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.40 - 6.40i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.03 - 3.99i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (-0.594 + 3.75i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-4.27 - 1.38i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 4.79i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-5.40 - 5.40i)T + 67iT^{2} \) |
| 71 | \( 1 + (7.19 + 5.23i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (10.1 - 5.17i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (4.69 - 3.40i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (0.200 + 1.26i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 10.5iT - 89T^{2} \) |
| 97 | \( 1 + (-6.92 - 1.09i)T + (92.2 + 29.9i)T^{2} \) |
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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
−13.39613915819476182465065425003, −12.98166557771408129014006470779, −11.34461536921642664557251371942, −10.26110814688213875955804758520, −9.351558194908334149175021965352, −8.415950423220455254485517776553, −7.49068004651916825060502158341, −4.67470490088629416335268569469, −3.93922888906942522304530318022, −2.57309381809814776985942713331,
2.70617374619830358340131518602, 3.90285356893142236007410683521, 6.21754140793733964930528817185, 7.35232005705063051714491020709, 8.230995710212933227498246884960, 8.798643744777629353719016273325, 10.35491314810287256936560450323, 12.15024747678816999733414162072, 12.94094920648136935817235079508, 13.98195954918260461519820594625
