| 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
−11.35337635410398057803778531414, −10.50403351773340395393061426727, −9.877784272480918581169183359778, −8.990804662561100508484876945535, −8.132926109964558021553070416917, −7.19875513700441764249433480563, −5.88689647891834501647123125427, −4.88112144170358287409169555860, −3.51306862946926595578816879310, −2.13104643515048270328639315970,
0.837443845706437985596845036752, 2.25936344599167431064747569560, 3.59192359598209276680384063153, 5.33353953963057264739781476701, 6.52224317385352767895608769308, 7.51421800898190508430942455448, 8.267291937567018289638273057402, 9.017861667586412636723405441824, 10.03734902210329709638294483327, 11.13408810198561253834652389182
