| L(s) = 1 | + (−1.16 + 1.62i)2-s + (−2.84 − 0.962i)3-s + (−1.29 − 3.78i)4-s + (1.39 + 7.88i)5-s + (4.86 − 3.50i)6-s + (−0.301 − 0.253i)7-s + (7.66 + 2.29i)8-s + (7.14 + 5.46i)9-s + (−14.4 − 6.90i)10-s + (−0.879 + 4.98i)11-s + (0.0401 + 11.9i)12-s + (1.67 − 4.59i)13-s + (0.762 − 0.196i)14-s + (3.63 − 23.7i)15-s + (−12.6 + 9.80i)16-s + (−18.3 + 10.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.581 + 0.813i)2-s + (−0.947 − 0.320i)3-s + (−0.323 − 0.946i)4-s + (0.278 + 1.57i)5-s + (0.811 − 0.584i)6-s + (−0.0430 − 0.0361i)7-s + (0.958 + 0.286i)8-s + (0.794 + 0.607i)9-s + (−1.44 − 0.690i)10-s + (−0.0799 + 0.453i)11-s + (0.00334 + 0.999i)12-s + (0.128 − 0.353i)13-s + (0.0544 − 0.0140i)14-s + (0.242 − 1.58i)15-s + (−0.790 + 0.612i)16-s + (−1.07 + 0.622i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.830 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0835668 - 0.274271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0835668 - 0.274271i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 - 1.62i)T \) |
| 3 | \( 1 + (2.84 + 0.962i)T \) |
| good | 5 | \( 1 + (-1.39 - 7.88i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (0.301 + 0.253i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (0.879 - 4.98i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-1.67 + 4.59i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (18.3 - 10.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (16.6 + 9.61i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (20.1 + 24.0i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-16.8 + 6.13i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (29.5 - 24.8i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (3.58 - 2.06i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (13.2 - 36.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-27.5 - 4.85i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (-44.7 + 53.2i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 105.T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-15.4 - 87.8i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (10.7 - 12.8i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (0.119 - 0.328i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (39.5 - 22.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (48.4 - 83.9i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (40.3 - 14.6i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-54.5 + 19.8i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-19.5 - 11.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-21.9 + 124. i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−12.74066240218787771837827887403, −11.28213981595945492599143041734, −10.55501923121105650119827808390, −10.09840136696718157268947809279, −8.530070913047629664658042265118, −7.25960083977287292127268984362, −6.61478568062599297399894742699, −5.94271125045942052309704585924, −4.45845816150893966047345262085, −2.13650654073426197751871594372,
0.21283449798358445428163218921, 1.69214805132390904392679507609, 4.00363480760916470132508209988, 4.88958233398703860742048930004, 6.13543692662612525801328833322, 7.76533857364728451467437564444, 9.005676279664459713829305036375, 9.436882286661242514567733724794, 10.65335996812516205830405379095, 11.48184444074674745440420475624
