| L(s) = 1 | + (−0.547 − 0.947i)2-s + (−6.76 − 5.94i)3-s + (7.40 − 12.8i)4-s + (−17.4 + 17.8i)5-s + (−1.93 + 9.65i)6-s + (−16.9 + 9.76i)7-s − 33.7·8-s + (10.4 + 80.3i)9-s + (26.5 + 6.79i)10-s + (−100. + 57.9i)11-s + (−126. + 42.6i)12-s + (−158. − 91.6i)13-s + (18.5 + 10.6i)14-s + (224. − 16.9i)15-s + (−99.9 − 173. i)16-s + 89.0·17-s + ⋯ |
| L(s) = 1 | + (−0.136 − 0.236i)2-s + (−0.751 − 0.660i)3-s + (0.462 − 0.801i)4-s + (−0.699 + 0.714i)5-s + (−0.0536 + 0.268i)6-s + (−0.345 + 0.199i)7-s − 0.526·8-s + (0.128 + 0.991i)9-s + (0.265 + 0.0679i)10-s + (−0.829 + 0.479i)11-s + (−0.876 + 0.296i)12-s + (−0.939 − 0.542i)13-s + (0.0944 + 0.0545i)14-s + (0.997 − 0.0751i)15-s + (−0.390 − 0.676i)16-s + 0.308·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.891 - 0.453i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0477427 + 0.198905i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0477427 + 0.198905i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.76 + 5.94i)T \) |
| 5 | \( 1 + (17.4 - 17.8i)T \) |
| good | 2 | \( 1 + (0.547 + 0.947i)T + (-8 + 13.8i)T^{2} \) |
| 7 | \( 1 + (16.9 - 9.76i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (100. - 57.9i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (158. + 91.6i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 - 89.0T + 8.35e4T^{2} \) |
| 19 | \( 1 + 52.2T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-232. + 401. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (1.25e3 - 724. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-767. + 1.33e3i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 641. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.19e3 + 690. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.77e3 - 1.02e3i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (589. + 1.02e3i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + 64.4T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.83e3 - 1.63e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-929. - 1.60e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-7.25e3 - 4.19e3i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.56e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 8.07e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-4.32e3 - 7.49e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.67e3 - 2.89e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.01e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.18e4 - 6.85e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.66555804158454250371730475810, −12.93967137976700955396972231917, −11.90268545814671255249401257179, −10.87040319542525864287176860042, −9.993201684389105912813893057787, −7.70638445083581602291320434853, −6.66765822504706115531857632515, −5.26352216552128522286112929855, −2.47899518473039174545151438567, −0.13766854309271920467703749520,
3.54955726160033275307419131708, 5.13224347636752261801949266311, 6.91587502437810273434501066477, 8.251137001504662875767548636739, 9.679527967170745251477659894883, 11.24046182335728883708904703154, 12.05961372983705421067083644839, 13.08841757291749545753733488308, 15.14796467458005158163812154024, 15.97391592590527863654375101942
