| L(s) = 1 | + (−0.388 − 0.411i)2-s + (0.0976 − 1.67i)4-s + (1.51 − 0.177i)5-s + (−1.38 − 0.695i)7-s + (−1.59 + 1.33i)8-s + (−0.663 − 0.556i)10-s + (0.219 + 0.509i)11-s + (−2.50 − 0.594i)13-s + (0.251 + 0.840i)14-s + (−2.16 − 0.253i)16-s + (0.994 − 5.64i)17-s + (−1.11 − 6.29i)19-s + (−0.149 − 2.56i)20-s + (0.124 − 0.288i)22-s + (4.93 − 2.47i)23-s + ⋯ |
| L(s) = 1 | + (−0.274 − 0.291i)2-s + (0.0488 − 0.838i)4-s + (0.679 − 0.0794i)5-s + (−0.523 − 0.263i)7-s + (−0.564 + 0.473i)8-s + (−0.209 − 0.176i)10-s + (0.0662 + 0.153i)11-s + (−0.696 − 0.164i)13-s + (0.0672 + 0.224i)14-s + (−0.541 − 0.0632i)16-s + (0.241 − 1.36i)17-s + (−0.254 − 1.44i)19-s + (−0.0334 − 0.573i)20-s + (0.0265 − 0.0614i)22-s + (1.02 − 0.516i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.295322 - 0.958997i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.295322 - 0.958997i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.388 + 0.411i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.51 + 0.177i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (1.38 + 0.695i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-0.219 - 0.509i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (2.50 + 0.594i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.994 + 5.64i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (1.11 + 6.29i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-4.93 + 2.47i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (1.93 - 6.46i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (0.637 + 0.419i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (7.41 - 2.69i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (0.471 - 0.500i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-4.70 + 6.32i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (4.72 - 3.10i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-5.99 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.926 - 2.14i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (0.637 + 10.9i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-2.89 - 9.65i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (5.17 + 4.34i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.85 + 4.91i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.31 - 3.51i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-6.22 - 6.59i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (10.4 - 8.73i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-6.78 - 0.793i)T + (94.3 + 22.3i)T^{2} \) |
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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
−9.910530495862456284413212706740, −9.423428341394097929031703156186, −8.721805411474881125385787363354, −7.12138656548894512893648833940, −6.68427624636663712427164388649, −5.35663962551562380947373356219, −4.90136489614650836943868673603, −3.10192205237228171546291509962, −2.07696851557238327884289149685, −0.53824544660030241524676494290,
1.96563342045343359352065861423, 3.21549850489793365482135341539, 4.15749187175997130213344500231, 5.72061149175489191163847362106, 6.29277660843795297711497724654, 7.37054392475192816103445248665, 8.121083513739275686938680918534, 9.038379126415530555898244762633, 9.747158715388193513240050858244, 10.52834790856620513180497338839
