| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.592 − 1.62i)3-s + (0.766 + 0.642i)4-s + (0.152 − 0.866i)5-s + (−1.11 + 1.32i)6-s + (0.766 − 0.642i)7-s + (−0.500 − 0.866i)8-s + (−2.29 − 1.92i)9-s + (−0.439 + 0.761i)10-s + (−0.358 − 2.03i)11-s + (1.5 − 0.866i)12-s + (−0.326 + 0.118i)13-s + (−0.939 + 0.342i)14-s + (−1.31 − 0.761i)15-s + (0.173 + 0.984i)16-s + (1.70 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.342 − 0.939i)3-s + (0.383 + 0.321i)4-s + (0.0682 − 0.387i)5-s + (−0.454 + 0.541i)6-s + (0.289 − 0.242i)7-s + (−0.176 − 0.306i)8-s + (−0.766 − 0.642i)9-s + (−0.139 + 0.240i)10-s + (−0.108 − 0.612i)11-s + (0.433 − 0.249i)12-s + (−0.0905 + 0.0329i)13-s + (−0.251 + 0.0914i)14-s + (−0.340 − 0.196i)15-s + (0.0434 + 0.246i)16-s + (0.413 − 0.716i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.597 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.462198 - 0.920313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.462198 - 0.920313i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.592 + 1.62i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
| good | 5 | \( 1 + (-0.152 + 0.866i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (0.358 + 2.03i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.326 - 0.118i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.70 + 2.95i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 1.76i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.36 + 3.66i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.78 + 0.650i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.03 + 1.70i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.226 - 0.392i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.06 - 1.84i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.336 + 1.90i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.04 + 1.71i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + (-0.177 + 1.00i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.62 - 2.20i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-10.0 + 3.66i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-2.87 + 4.97i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.20 - 10.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.33 - 2.30i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.41 - 1.24i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-9.38 - 16.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.24 + 7.05i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.13408810198561253834652389182, −10.03734902210329709638294483327, −9.017861667586412636723405441824, −8.267291937567018289638273057402, −7.51421800898190508430942455448, −6.52224317385352767895608769308, −5.33353953963057264739781476701, −3.59192359598209276680384063153, −2.25936344599167431064747569560, −0.837443845706437985596845036752,
2.13104643515048270328639315970, 3.51306862946926595578816879310, 4.88112144170358287409169555860, 5.88689647891834501647123125427, 7.19875513700441764249433480563, 8.132926109964558021553070416917, 8.990804662561100508484876945535, 9.877784272480918581169183359778, 10.50403351773340395393061426727, 11.35337635410398057803778531414
