| L(s) = 1 | + (−1.94 − 0.468i)2-s + (4.04 − 2.06i)3-s + (3.56 + 1.82i)4-s + (2.33 + 4.42i)5-s + (−8.83 + 2.11i)6-s + (3.03 − 5.94i)7-s + (−6.06 − 5.21i)8-s + (6.82 − 9.40i)9-s + (−2.46 − 9.69i)10-s + (2.74 − 10.6i)11-s + (18.1 + 0.0352i)12-s + (3.87 + 24.4i)13-s + (−8.68 + 10.1i)14-s + (18.5 + 13.0i)15-s + (9.35 + 12.9i)16-s + (2.85 − 18.0i)17-s + ⋯ |
| L(s) = 1 | + (−0.972 − 0.234i)2-s + (1.34 − 0.687i)3-s + (0.890 + 0.455i)4-s + (0.466 + 0.884i)5-s + (−1.47 + 0.351i)6-s + (0.432 − 0.849i)7-s + (−0.758 − 0.651i)8-s + (0.758 − 1.04i)9-s + (−0.246 − 0.969i)10-s + (0.249 − 0.968i)11-s + (1.51 + 0.00293i)12-s + (0.297 + 1.88i)13-s + (−0.620 + 0.724i)14-s + (1.23 + 0.872i)15-s + (0.584 + 0.811i)16-s + (0.167 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 220 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.67401 - 0.609585i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67401 - 0.609585i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.94 + 0.468i)T \) |
| 5 | \( 1 + (-2.33 - 4.42i)T \) |
| 11 | \( 1 + (-2.74 + 10.6i)T \) |
| good | 3 | \( 1 + (-4.04 + 2.06i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-3.03 + 5.94i)T + (-28.8 - 39.6i)T^{2} \) |
| 13 | \( 1 + (-3.87 - 24.4i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-2.85 + 18.0i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (10.7 + 3.50i)T + (292. + 212. i)T^{2} \) |
| 23 | \( 1 + (-20.8 - 20.8i)T + 529iT^{2} \) |
| 29 | \( 1 + (9.88 + 30.4i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-8.80 + 12.1i)T + (-296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-6.94 + 13.6i)T + (-804. - 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-9.85 - 3.20i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (14.2 - 14.2i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (8.10 + 15.9i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (69.7 - 11.0i)T + (2.67e3 - 868. i)T^{2} \) |
| 59 | \( 1 + (24.7 + 76.2i)T + (-2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (22.3 + 30.8i)T + (-1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (15.3 - 15.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + (-12.2 - 16.8i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (36.2 - 71.1i)T + (-3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (58.2 - 80.1i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (16.4 - 103. i)T + (-6.55e3 - 2.12e3i)T^{2} \) |
| 89 | \( 1 + 69.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-22.7 - 143. i)T + (-8.94e3 + 2.90e3i)T^{2} \) |
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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
−11.48352369577881022341338933741, −11.09919015953136227966554957250, −9.624694074592733122497358094490, −9.093295533164850842484273945057, −7.966910453271125694105457573203, −7.16642047394340546819569703529, −6.44879846963166281844477312299, −3.77521848768720103556202588685, −2.62046017888274818418626728230, −1.46305290132270866806186471019,
1.69454096396767394175808387714, 2.97394293738397188842132653410, 4.80375317448095740809673496352, 5.97768539936847242031724086561, 7.72900868140672034907467909018, 8.594167020742850009764317038977, 8.885406265517275073700714800932, 10.01106921365173659953892459234, 10.60746794196376060904740847285, 12.30668481603257054415636618362
