| L(s) = 1 | + (1.47 + 1.35i)2-s + (−0.579 + 2.94i)3-s + (0.322 + 3.98i)4-s + (9.22 − 3.35i)5-s + (−4.84 + 3.54i)6-s + (−4.51 + 0.795i)7-s + (−4.93 + 6.29i)8-s + (−8.32 − 3.41i)9-s + (18.1 + 7.57i)10-s + (2.67 − 7.34i)11-s + (−11.9 − 1.36i)12-s + (−5.33 − 4.47i)13-s + (−7.71 − 4.94i)14-s + (4.53 + 29.0i)15-s + (−15.7 + 2.57i)16-s + (−7.51 + 13.0i)17-s + ⋯ |
| L(s) = 1 | + (0.735 + 0.677i)2-s + (−0.193 + 0.981i)3-s + (0.0806 + 0.996i)4-s + (1.84 − 0.671i)5-s + (−0.807 + 0.590i)6-s + (−0.644 + 0.113i)7-s + (−0.616 + 0.787i)8-s + (−0.925 − 0.379i)9-s + (1.81 + 0.757i)10-s + (0.243 − 0.668i)11-s + (−0.993 − 0.113i)12-s + (−0.410 − 0.344i)13-s + (−0.550 − 0.353i)14-s + (0.302 + 1.93i)15-s + (−0.986 + 0.160i)16-s + (−0.441 + 0.765i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0791 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.43940 + 1.55819i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43940 + 1.55819i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.47 - 1.35i)T \) |
| 3 | \( 1 + (0.579 - 2.94i)T \) |
| good | 5 | \( 1 + (-9.22 + 3.35i)T + (19.1 - 16.0i)T^{2} \) |
| 7 | \( 1 + (4.51 - 0.795i)T + (46.0 - 16.7i)T^{2} \) |
| 11 | \( 1 + (-2.67 + 7.34i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (5.33 + 4.47i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (7.51 - 13.0i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-18.4 + 10.6i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (16.6 + 2.94i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (-14.8 + 12.4i)T + (146. - 828. i)T^{2} \) |
| 31 | \( 1 + (-14.8 - 2.61i)T + (903. + 328. i)T^{2} \) |
| 37 | \( 1 + (14.9 - 25.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-29.3 - 24.6i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-14.2 + 39.2i)T + (-1.41e3 - 1.18e3i)T^{2} \) |
| 47 | \( 1 + (60.2 - 10.6i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 + 24.8T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-31.1 - 85.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-4.84 - 27.4i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (25.2 - 30.0i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (34.2 + 19.7i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (41.3 + 71.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.3 + 12.3i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (55.6 + 66.3i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (5.93 + 10.2i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-47.7 - 17.3i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−13.76321650192047231208711137449, −13.06276914310341922826783413048, −11.88704561920764875813685526492, −10.33245705938281738403930705323, −9.410152141314890774069857280322, −8.536410199529151977868503872454, −6.34944188302643208606539861588, −5.73559529130241030785037035288, −4.66046214830529369613645417748, −2.89602139806169582954595131735,
1.74605813513729262222527148181, 2.85472243934310579748493659515, 5.24757562129387490420437855880, 6.29914725894258552058127128638, 7.00472388335742137677719442172, 9.430605003586487971013874739755, 10.00709070064082104268225713487, 11.25492039115211229215426774961, 12.41289432379177091303693054461, 13.19472038901187092580211956901
