| L(s) = 1 | + (0.361 + 0.839i)2-s + (0.799 − 0.847i)4-s + (0.221 − 3.80i)5-s + (−0.706 + 0.167i)7-s + (2.71 + 0.989i)8-s + (3.27 − 1.19i)10-s + (2.24 − 1.47i)11-s + (−4.57 + 0.534i)13-s + (−0.396 − 0.532i)14-s + (0.0182 + 0.313i)16-s + (−0.692 − 0.581i)17-s + (−1.12 + 0.940i)19-s + (−3.04 − 3.22i)20-s + (2.05 + 1.35i)22-s + (3.79 + 0.899i)23-s + ⋯ |
| L(s) = 1 | + (0.255 + 0.593i)2-s + (0.399 − 0.423i)4-s + (0.0990 − 1.70i)5-s + (−0.267 + 0.0633i)7-s + (0.960 + 0.349i)8-s + (1.03 − 0.376i)10-s + (0.678 − 0.446i)11-s + (−1.26 + 0.148i)13-s + (−0.105 − 0.142i)14-s + (0.00455 + 0.0782i)16-s + (−0.168 − 0.140i)17-s + (−0.257 + 0.215i)19-s + (−0.680 − 0.721i)20-s + (0.438 + 0.288i)22-s + (0.791 + 0.187i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.65301 - 0.934031i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65301 - 0.934031i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-0.361 - 0.839i)T + (-1.37 + 1.45i)T^{2} \) |
| 5 | \( 1 + (-0.221 + 3.80i)T + (-4.96 - 0.580i)T^{2} \) |
| 7 | \( 1 + (0.706 - 0.167i)T + (6.25 - 3.14i)T^{2} \) |
| 11 | \( 1 + (-2.24 + 1.47i)T + (4.35 - 10.1i)T^{2} \) |
| 13 | \( 1 + (4.57 - 0.534i)T + (12.6 - 2.99i)T^{2} \) |
| 17 | \( 1 + (0.692 + 0.581i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (1.12 - 0.940i)T + (3.29 - 18.7i)T^{2} \) |
| 23 | \( 1 + (-3.79 - 0.899i)T + (20.5 + 10.3i)T^{2} \) |
| 29 | \( 1 + (-3.06 + 4.12i)T + (-8.31 - 27.7i)T^{2} \) |
| 31 | \( 1 + (2.86 + 9.56i)T + (-25.9 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.348 - 1.97i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (2.59 - 6.02i)T + (-28.1 - 29.8i)T^{2} \) |
| 43 | \( 1 + (-6.51 - 3.27i)T + (25.6 + 34.4i)T^{2} \) |
| 47 | \( 1 + (-1.28 + 4.28i)T + (-39.2 - 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.43 + 5.95i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.590 - 0.388i)T + (23.3 + 54.1i)T^{2} \) |
| 61 | \( 1 + (-1.83 - 1.94i)T + (-3.54 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-8.91 - 11.9i)T + (-19.2 + 64.1i)T^{2} \) |
| 71 | \( 1 + (9.19 - 3.34i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-15.0 - 5.48i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-1.89 - 4.38i)T + (-54.2 + 57.4i)T^{2} \) |
| 83 | \( 1 + (2.86 + 6.65i)T + (-56.9 + 60.3i)T^{2} \) |
| 89 | \( 1 + (-7.00 - 2.54i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-0.258 - 4.44i)T + (-96.3 + 11.2i)T^{2} \) |
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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
−9.924220906145782488934106458873, −9.447115249103703612688698977451, −8.469559237609708870863785775430, −7.63714663806096520574098878689, −6.59942899521867020346807648629, −5.70985624838232379642823757085, −4.95222812117519027575403089473, −4.17127942388415133720115876650, −2.26395716908641392786905355487, −0.916170980047808967567621935786,
2.03161399959911737212302641996, 2.91951269811420191172383790334, 3.65893524796707417662731600552, 4.91026670728951652488310743013, 6.49585469159696679567416759081, 7.00306621812557112006224062470, 7.55451172791414309464657874867, 8.998192195814781502775239103803, 10.06848704736106145338261835710, 10.65841653007622553608675119419
