| L(s) = 1 | + (0.891 + 0.453i)2-s + (−1.69 − 0.333i)3-s + (0.587 + 0.809i)4-s + (1.62 − 1.53i)5-s + (−1.36 − 1.06i)6-s + (2.58 + 2.58i)7-s + (0.156 + 0.987i)8-s + (2.77 + 1.13i)9-s + (2.14 − 0.633i)10-s + (1.45 − 0.473i)11-s + (−0.729 − 1.57i)12-s + (−2.28 − 4.48i)13-s + (1.12 + 3.47i)14-s + (−3.27 + 2.07i)15-s + (−0.309 + 0.951i)16-s + (−5.09 + 0.806i)17-s + ⋯ |
| L(s) = 1 | + (0.630 + 0.321i)2-s + (−0.981 − 0.192i)3-s + (0.293 + 0.404i)4-s + (0.725 − 0.687i)5-s + (−0.556 − 0.436i)6-s + (0.976 + 0.976i)7-s + (0.0553 + 0.349i)8-s + (0.925 + 0.378i)9-s + (0.678 − 0.200i)10-s + (0.439 − 0.142i)11-s + (−0.210 − 0.453i)12-s + (−0.633 − 1.24i)13-s + (0.301 + 0.928i)14-s + (−0.844 + 0.535i)15-s + (−0.0772 + 0.237i)16-s + (−1.23 + 0.195i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 - 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.36023 + 0.196253i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.36023 + 0.196253i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.891 - 0.453i)T \) |
| 3 | \( 1 + (1.69 + 0.333i)T \) |
| 5 | \( 1 + (-1.62 + 1.53i)T \) |
| good | 7 | \( 1 + (-2.58 - 2.58i)T + 7iT^{2} \) |
| 11 | \( 1 + (-1.45 + 0.473i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (2.28 + 4.48i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (5.09 - 0.806i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.27 - 1.75i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (2.88 - 5.66i)T + (-13.5 - 18.6i)T^{2} \) |
| 29 | \( 1 + (-5.64 + 4.10i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.95 + 4.32i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (6.84 - 3.48i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-2.85 - 0.929i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-3.24 + 3.24i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.446 - 2.81i)T + (-44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (1.02 + 0.161i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (2.10 - 6.48i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.78 - 5.49i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (0.527 + 3.33i)T + (-63.7 + 20.7i)T^{2} \) |
| 71 | \( 1 + (2.18 + 3.00i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.531 + 0.270i)T + (42.9 + 59.0i)T^{2} \) |
| 79 | \( 1 + (0.782 + 1.07i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (0.432 + 2.73i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (1.98 + 6.12i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-13.5 - 2.14i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.93083093342037960440618788324, −12.17908783397502579080644970807, −11.40198671193840130071121909519, −10.18456147414496255341574694789, −8.798162500222379568136441871032, −7.67203594027782225306411489197, −6.10787092990314121595657397752, −5.47828789238426623320928865926, −4.52809015085892531742048115183, −1.99938115650086871880136276126,
1.90458393620085550536203445704, 4.18529424870313336145242282259, 4.98590758467702972680347300231, 6.57134579400241466087583084182, 7.03422864685533135225323723646, 9.187603266151546561996804354741, 10.46367414436682323857869165360, 10.92175438333575275390971669610, 11.81812023777220493861144120542, 12.91030423681915822772791940265
