| L(s) = 1 | + (−0.0763 + 0.482i)2-s + (−1.01 − 0.517i)3-s + (1.67 + 0.544i)4-s + (1.47 + 1.67i)5-s + (0.327 − 0.450i)6-s + (−1.32 − 2.59i)7-s + (−0.833 + 1.63i)8-s + (−1.00 − 1.37i)9-s + (−0.922 + 0.583i)10-s + (−3.15 + 1.00i)11-s + (−1.41 − 1.41i)12-s + (−2.89 − 0.457i)13-s + (1.35 − 0.440i)14-s + (−0.630 − 2.46i)15-s + (2.12 + 1.54i)16-s + (5.20 − 0.824i)17-s + ⋯ |
| L(s) = 1 | + (−0.0540 + 0.341i)2-s + (−0.586 − 0.298i)3-s + (0.837 + 0.272i)4-s + (0.660 + 0.750i)5-s + (0.133 − 0.183i)6-s + (−0.500 − 0.982i)7-s + (−0.294 + 0.578i)8-s + (−0.333 − 0.458i)9-s + (−0.291 + 0.184i)10-s + (−0.952 + 0.303i)11-s + (−0.409 − 0.409i)12-s + (−0.801 − 0.126i)13-s + (0.362 − 0.117i)14-s + (−0.162 − 0.637i)15-s + (0.531 + 0.385i)16-s + (1.26 − 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.808898 + 0.162501i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.808898 + 0.162501i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.47 - 1.67i)T \) |
| 11 | \( 1 + (3.15 - 1.00i)T \) |
| good | 2 | \( 1 + (0.0763 - 0.482i)T + (-1.90 - 0.618i)T^{2} \) |
| 3 | \( 1 + (1.01 + 0.517i)T + (1.76 + 2.42i)T^{2} \) |
| 7 | \( 1 + (1.32 + 2.59i)T + (-4.11 + 5.66i)T^{2} \) |
| 13 | \( 1 + (2.89 + 0.457i)T + (12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (-5.20 + 0.824i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (1.26 + 3.89i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.12 - 2.12i)T - 23iT^{2} \) |
| 29 | \( 1 + (0.817 - 2.51i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.45 + 3.96i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.03 + 0.528i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (3.38 - 1.09i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-5.07 - 5.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (-1.67 + 3.28i)T + (-27.6 - 38.0i)T^{2} \) |
| 53 | \( 1 + (0.231 - 1.45i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-1.52 - 0.496i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.07 - 5.61i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-1.31 - 1.31i)T + 67iT^{2} \) |
| 71 | \( 1 + (2.32 + 1.68i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-13.1 + 6.71i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (12.3 - 8.96i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (1.04 + 6.60i)T + (-78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + 11.1iT - 89T^{2} \) |
| 97 | \( 1 + (9.94 + 1.57i)T + (92.2 + 29.9i)T^{2} \) |
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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
−15.43675659763463737610677871224, −14.42069015395907529765828125579, −13.12033998132205128081335120750, −11.94152306563603798569516178674, −10.77993266581096415608740489323, −9.841705343875240920107397713405, −7.60323098552189778215801356610, −6.80631181470709237066262699731, −5.65392656480380699789271697872, −2.91867118039357268412270919164,
2.48541596424522685325158092750, 5.32658686580482178258906802945, 6.05772449389515383177921616766, 8.128712986140020156192546319857, 9.781490981092751113106528492603, 10.50449462914034199524372451343, 12.00407844214981687080053356005, 12.54990291665148342613343439403, 14.15893607534245347908240112438, 15.58277568994631727824305075090
