| L(s) = 1 | + (−1.95 + 0.424i)2-s + (0.0189 + 2.99i)3-s + (3.63 − 1.65i)4-s + (−1.07 − 6.12i)5-s + (−1.30 − 5.85i)6-s + (6.40 − 7.62i)7-s + (−6.41 + 4.78i)8-s + (−8.99 + 0.113i)9-s + (4.70 + 11.5i)10-s + (12.4 + 2.19i)11-s + (5.04 + 10.8i)12-s + (12.1 + 4.42i)13-s + (−9.27 + 17.6i)14-s + (18.3 − 3.35i)15-s + (10.4 − 12.0i)16-s + (−9.19 − 15.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.977 + 0.212i)2-s + (0.00631 + 0.999i)3-s + (0.909 − 0.414i)4-s + (−0.215 − 1.22i)5-s + (−0.218 − 0.975i)6-s + (0.914 − 1.08i)7-s + (−0.801 + 0.598i)8-s + (−0.999 + 0.0126i)9-s + (0.470 + 1.15i)10-s + (1.13 + 0.199i)11-s + (0.420 + 0.907i)12-s + (0.934 + 0.340i)13-s + (−0.662 + 1.25i)14-s + (1.22 − 0.223i)15-s + (0.656 − 0.754i)16-s + (−0.541 − 0.937i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.952375 - 0.0645411i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.952375 - 0.0645411i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.95 - 0.424i)T \) |
| 3 | \( 1 + (-0.0189 - 2.99i)T \) |
| good | 5 | \( 1 + (1.07 + 6.12i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-6.40 + 7.62i)T + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-12.4 - 2.19i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (-12.1 - 4.42i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (9.19 + 15.9i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4.75 - 2.74i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.65 - 7.92i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-9.36 + 3.40i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-20.5 - 24.4i)T + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (28.6 + 49.6i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (8.81 + 3.20i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (65.6 + 11.5i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (19.6 - 23.3i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 59.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-102. + 18.0i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (24.2 + 20.3i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (33.8 - 93.0i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (72.6 - 41.9i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (45.9 - 79.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-25.9 - 71.1i)T + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (9.07 + 24.9i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (53.2 - 92.3i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-0.0639 + 0.362i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−13.74616125078308435104902366936, −11.84010042144193490098648288990, −11.22618394332378545020683389151, −10.08360047670934431489623541786, −8.970698055574537562866863575722, −8.425027846264408234249501690997, −6.94629201328257272725827477304, −5.18757677722753527084241563711, −4.03266381210211581259843350877, −1.13472093811303062992206473952,
1.70096063124499945436100260475, 3.13237203986203802939039567327, 6.11147624637983319402430840841, 6.84133616107629701108227794063, 8.218431353058833362691884090843, 8.769122219456364237678798912487, 10.52395732390398726170867422647, 11.54329152574219870505925330315, 11.87068067572028446097464342181, 13.39770540583695105945623165551
