| L(s) = 1 | + (0.137 + 1.84i)2-s + (−1.49 − 0.462i)3-s + (−1.39 + 0.209i)4-s + (−1.02 + 0.946i)5-s + (0.644 − 2.82i)6-s + (0.242 + 1.06i)8-s + (−0.446 − 0.304i)9-s + (−1.88 − 1.74i)10-s + (−2.38 + 1.62i)11-s + (2.18 + 0.329i)12-s + (−4.03 − 1.94i)13-s + (1.96 − 0.947i)15-s + (−4.61 + 1.42i)16-s + (−0.176 − 0.449i)17-s + (0.498 − 0.863i)18-s + (−1.90 − 3.30i)19-s + ⋯ |
| L(s) = 1 | + (0.0975 + 1.30i)2-s + (−0.865 − 0.266i)3-s + (−0.696 + 0.104i)4-s + (−0.456 + 0.423i)5-s + (0.263 − 1.15i)6-s + (0.0859 + 0.376i)8-s + (−0.148 − 0.101i)9-s + (−0.595 − 0.552i)10-s + (−0.718 + 0.489i)11-s + (0.630 + 0.0950i)12-s + (−1.12 − 0.539i)13-s + (0.507 − 0.244i)15-s + (−1.15 + 0.356i)16-s + (−0.0427 − 0.108i)17-s + (0.117 − 0.203i)18-s + (−0.437 − 0.757i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.491 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.125021 - 0.214117i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.125021 - 0.214117i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-0.137 - 1.84i)T + (-1.97 + 0.298i)T^{2} \) |
| 3 | \( 1 + (1.49 + 0.462i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (1.02 - 0.946i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (2.38 - 1.62i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (4.03 + 1.94i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (0.176 + 0.449i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (1.90 + 3.30i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.48 + 6.33i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (-1.45 + 1.81i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (3.94 - 6.82i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (9.48 + 1.42i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (-1.56 - 6.84i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-0.546 + 2.39i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.836 - 11.1i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (5.30 - 0.799i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-2.74 - 2.54i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-0.0400 - 0.00602i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (0.534 - 0.926i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.21 - 5.28i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-0.613 + 8.18i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.91 - 11.9i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.465 + 0.224i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.648 + 0.442i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 9.61T + 97T^{2} \) |
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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
−12.24547457288131126631049641010, −11.13270459901950951294262348508, −10.54539390687550934041318463224, −9.074068648112886191963578509108, −7.983556948411467188275694761380, −7.10973704717969913084618349753, −6.59110766024653724263702881319, −5.37723925980010792954940338190, −4.76816456275483899753470929777, −2.74354714195193281970997446711,
0.17227754186270865656837379577, 2.13125831726073834938837752789, 3.57070143110947802255652838169, 4.70528836011067356056825131083, 5.60135726839219267027933424137, 7.04487066471729040024922472845, 8.242523938772142843516294090432, 9.444726522485869556578834381780, 10.36599690463710262842718355209, 10.98612071685003219911534623023
