| L(s) = 1 | + (−0.316 − 0.548i)2-s + (−4.90 + 7.54i)3-s + (7.79 − 13.5i)4-s + (−24.0 − 6.92i)5-s + (5.68 + 0.303i)6-s + (73.2 − 42.2i)7-s − 20.0·8-s + (−32.8 − 74.0i)9-s + (3.80 + 15.3i)10-s + (148. − 85.7i)11-s + (63.6 + 125. i)12-s + (−127. − 73.4i)13-s + (−46.3 − 26.7i)14-s + (170. − 147. i)15-s + (−118. − 205. i)16-s − 273.·17-s + ⋯ |
| L(s) = 1 | + (−0.0791 − 0.137i)2-s + (−0.545 + 0.838i)3-s + (0.487 − 0.844i)4-s + (−0.960 − 0.277i)5-s + (0.158 + 0.00843i)6-s + (1.49 − 0.862i)7-s − 0.312·8-s + (−0.404 − 0.914i)9-s + (0.0380 + 0.153i)10-s + (1.22 − 0.708i)11-s + (0.441 + 0.869i)12-s + (−0.752 − 0.434i)13-s + (−0.236 − 0.136i)14-s + (0.756 − 0.654i)15-s + (−0.462 − 0.801i)16-s − 0.946·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.960633 - 0.700427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.960633 - 0.700427i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.90 - 7.54i)T \) |
| 5 | \( 1 + (24.0 + 6.92i)T \) |
| good | 2 | \( 1 + (0.316 + 0.548i)T + (-8 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-73.2 + 42.2i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-148. + 85.7i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (127. + 73.4i)T + (1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + 273.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 229.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (194. - 337. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-306. + 177. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (310. - 538. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 730. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (498. + 288. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-783. + 452. i)T + (1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (396. + 686. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 - 3.46e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-2.43e3 - 1.40e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-2.66e3 - 4.60e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (147. + 85.4i)T + (1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 - 2.79e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 5.86e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (-1.46e3 - 2.54e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (3.44e3 + 5.96e3i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + 1.03e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.21e4 - 7.02e3i)T + (4.42e7 - 7.66e7i)T^{2} \) |
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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
−14.91978344140237222910313283803, −14.16699981944638110972587105849, −11.74301956630363700087177639532, −11.37137593196246578231103373723, −10.33053584728937161757794031316, −8.802780070592511713158918707833, −7.13103823959496792131443385010, −5.30544695114236598917917413469, −4.07812609529349284452719177103, −0.882466714496700014475211878763,
2.12871547398196250959023679588, 4.57947059120509671191173669098, 6.67448079839478840943501380326, 7.64425536935768709792169980531, 8.655899840325581991666753428043, 11.27867423033379460572971346881, 11.77168727665550870381353324840, 12.44465090954933346214494522737, 14.34217374760464619747235834499, 15.29232611853383460248300043706
