| L(s) = 1 | + (−0.434 − 1.62i)2-s + (2.96 − 0.485i)3-s + (1.02 − 0.590i)4-s + (−2.94 + 4.03i)5-s + (−2.07 − 4.59i)6-s + (−0.710 − 2.65i)7-s + (−6.15 − 6.15i)8-s + (8.52 − 2.87i)9-s + (7.83 + 3.02i)10-s + (−7.26 + 12.5i)11-s + (2.73 − 2.24i)12-s + (−1.52 + 5.67i)13-s + (−3.99 + 2.30i)14-s + (−6.76 + 13.3i)15-s + (−4.94 + 8.56i)16-s + (5.66 − 5.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.217 − 0.810i)2-s + (0.986 − 0.161i)3-s + (0.255 − 0.147i)4-s + (−0.589 + 0.807i)5-s + (−0.345 − 0.765i)6-s + (−0.101 − 0.378i)7-s + (−0.768 − 0.768i)8-s + (0.947 − 0.319i)9-s + (0.783 + 0.302i)10-s + (−0.660 + 1.14i)11-s + (0.228 − 0.187i)12-s + (−0.117 + 0.436i)13-s + (−0.285 + 0.164i)14-s + (−0.451 + 0.892i)15-s + (−0.308 + 0.535i)16-s + (0.333 − 0.333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.511 + 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.10353 - 0.626997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10353 - 0.626997i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.96 + 0.485i)T \) |
| 5 | \( 1 + (2.94 - 4.03i)T \) |
| good | 2 | \( 1 + (0.434 + 1.62i)T + (-3.46 + 2i)T^{2} \) |
| 7 | \( 1 + (0.710 + 2.65i)T + (-42.4 + 24.5i)T^{2} \) |
| 11 | \( 1 + (7.26 - 12.5i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (1.52 - 5.67i)T + (-146. - 84.5i)T^{2} \) |
| 17 | \( 1 + (-5.66 + 5.66i)T - 289iT^{2} \) |
| 19 | \( 1 - 30.5iT - 361T^{2} \) |
| 23 | \( 1 + (-5.85 + 21.8i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + (-1.51 - 0.873i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (25.5 + 44.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-2.78 + 2.78i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (-16.5 - 28.6i)T + (-840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (38.2 - 10.2i)T + (1.60e3 - 924.5i)T^{2} \) |
| 47 | \( 1 + (11.0 + 41.1i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (4.77 + 4.77i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-69.8 + 40.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (7.61 - 13.1i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (61.8 + 16.5i)T + (3.88e3 + 2.24e3i)T^{2} \) |
| 71 | \( 1 - 45.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-92.0 - 92.0i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + (28.1 + 16.2i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-6.68 + 1.79i)T + (5.96e3 - 3.44e3i)T^{2} \) |
| 89 | \( 1 + 35.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-15.9 - 59.6i)T + (-8.14e3 + 4.70e3i)T^{2} \) |
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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
−15.10103140329302608908231456668, −14.50181241725310358303127735145, −12.89910858029448159604128060139, −11.86829770810831976591863577186, −10.43598773971949842101758922440, −9.721081723913005060010932232812, −7.86144157582365232834947204690, −6.80178996631623664776696804958, −3.81350218845994262986614949392, −2.29460159466485596522587284135,
3.12947983540622196733420259228, 5.33404819666225998516051077394, 7.30525753100478390061497439283, 8.345934098860635895607083724393, 9.072651353846206152449606743101, 11.05139692030497610169930618952, 12.51138804422536766902282992631, 13.63694819685009673418483043561, 15.12135166643613432394694658971, 15.73043063902938168974998004984
