| L(s) = 1 | + (0.259 + 0.275i)2-s + (−0.322 − 1.70i)3-s + (0.107 − 1.85i)4-s + (−1.27 + 0.148i)5-s + (0.384 − 0.530i)6-s + (1.70 + 0.857i)7-s + (1.11 − 0.938i)8-s + (−2.79 + 1.09i)9-s + (−0.371 − 0.312i)10-s + (1.93 + 4.48i)11-s + (−3.18 + 0.413i)12-s + (3.34 + 0.793i)13-s + (0.207 + 0.692i)14-s + (0.663 + 2.11i)15-s + (−3.13 − 0.366i)16-s + (0.0728 − 0.412i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.194i)2-s + (−0.185 − 0.982i)3-s + (0.0539 − 0.926i)4-s + (−0.569 + 0.0665i)5-s + (0.157 − 0.216i)6-s + (0.645 + 0.324i)7-s + (0.395 − 0.331i)8-s + (−0.930 + 0.365i)9-s + (−0.117 − 0.0986i)10-s + (0.582 + 1.35i)11-s + (−0.920 + 0.119i)12-s + (0.928 + 0.220i)13-s + (0.0554 + 0.185i)14-s + (0.171 + 0.547i)15-s + (−0.784 − 0.0917i)16-s + (0.0176 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.874812 - 0.433171i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.874812 - 0.433171i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.322 + 1.70i)T \) |
| good | 2 | \( 1 + (-0.259 - 0.275i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (1.27 - 0.148i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (-1.70 - 0.857i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-1.93 - 4.48i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (-3.34 - 0.793i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.0728 + 0.412i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (0.626 + 3.55i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (5.99 - 3.01i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-1.44 + 4.81i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-5.54 - 3.64i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-0.465 + 0.169i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-4.16 + 4.41i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (6.92 - 9.30i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (5.06 - 3.33i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-2.95 + 5.10i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.23 + 5.17i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.161 - 2.77i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-2.07 - 6.93i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (11.9 + 10.0i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (5.04 - 4.23i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (2.62 + 2.78i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (-3.32 - 3.52i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (7.97 - 6.69i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.14 - 0.251i)T + (94.3 + 22.3i)T^{2} \) |
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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
−14.22215875135161313944160586624, −13.28006667059237293768355894649, −11.87835531060008224664027907127, −11.32074564646154471386920530588, −9.790742593875732048047470426518, −8.294546716258683424721739727282, −7.08168920326359930500471613984, −6.03512718996633684943517999056, −4.56187894543284082678708233688, −1.76092800498733008326588397997,
3.47355894831640250403254915817, 4.29511485383354791600224903577, 6.07136904630790737898561641989, 8.083861829413223929873640103018, 8.624975573484237862577835163050, 10.41559780063554598574297468133, 11.40001664117603995699720999852, 11.98250274988202838958745375086, 13.58267305045831086682175528865, 14.44229368476028323732606301151
