| L(s) = 1 | + (−1.34 − 0.592i)2-s + (−1.01 − 0.595i)3-s + (0.103 + 0.113i)4-s + (2.74 − 3.15i)5-s + (1.00 + 1.40i)6-s + (3.67 + 2.91i)7-s + (0.861 + 2.56i)8-s + (−0.787 − 1.41i)9-s + (−5.56 + 2.61i)10-s + (1.76 − 0.637i)11-s + (−0.0371 − 0.175i)12-s + (4.44 + 2.21i)13-s + (−3.20 − 6.08i)14-s + (−4.66 + 1.56i)15-s + (0.395 − 4.26i)16-s + (−3.82 − 1.54i)17-s + ⋯ |
| L(s) = 1 | + (−0.949 − 0.419i)2-s + (−0.585 − 0.343i)3-s + (0.0515 + 0.0565i)4-s + (1.22 − 1.41i)5-s + (0.411 + 0.571i)6-s + (1.39 + 1.10i)7-s + (0.304 + 0.908i)8-s + (−0.262 − 0.471i)9-s + (−1.75 + 0.825i)10-s + (0.532 − 0.192i)11-s + (−0.0107 − 0.0508i)12-s + (1.23 + 0.614i)13-s + (−0.857 − 1.62i)14-s + (−1.20 + 0.404i)15-s + (0.0987 − 1.06i)16-s + (−0.926 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0196 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0196 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.621302 - 0.633641i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.621302 - 0.633641i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (3.82 + 1.54i)T \) |
| good | 2 | \( 1 + (1.34 + 0.592i)T + (1.34 + 1.47i)T^{2} \) |
| 3 | \( 1 + (1.01 + 0.595i)T + (1.45 + 2.62i)T^{2} \) |
| 5 | \( 1 + (-2.74 + 3.15i)T + (-0.690 - 4.95i)T^{2} \) |
| 7 | \( 1 + (-3.67 - 2.91i)T + (1.60 + 6.81i)T^{2} \) |
| 11 | \( 1 + (-1.76 + 0.637i)T + (8.46 - 7.02i)T^{2} \) |
| 13 | \( 1 + (-4.44 - 2.21i)T + (7.83 + 10.3i)T^{2} \) |
| 19 | \( 1 + (-0.790 - 1.79i)T + (-12.8 + 14.0i)T^{2} \) |
| 23 | \( 1 + (-2.66 + 3.35i)T + (-5.26 - 22.3i)T^{2} \) |
| 29 | \( 1 + (0.978 + 0.459i)T + (18.5 + 22.3i)T^{2} \) |
| 31 | \( 1 + (0.0263 - 0.380i)T + (-30.7 - 4.28i)T^{2} \) |
| 37 | \( 1 + (7.22 - 4.70i)T + (14.9 - 33.8i)T^{2} \) |
| 41 | \( 1 + (5.94 - 1.54i)T + (35.8 - 19.9i)T^{2} \) |
| 43 | \( 1 + (4.04 + 3.36i)T + (7.90 + 42.2i)T^{2} \) |
| 47 | \( 1 + (3.26 - 0.929i)T + (39.9 - 24.7i)T^{2} \) |
| 53 | \( 1 + (-8.06 + 4.49i)T + (27.9 - 45.0i)T^{2} \) |
| 59 | \( 1 + (1.31 - 5.59i)T + (-52.8 - 26.2i)T^{2} \) |
| 61 | \( 1 + (6.09 - 4.38i)T + (19.3 - 57.8i)T^{2} \) |
| 67 | \( 1 + (1.72 + 0.669i)T + (49.5 + 45.1i)T^{2} \) |
| 71 | \( 1 + (-0.115 - 0.993i)T + (-69.1 + 16.2i)T^{2} \) |
| 73 | \( 1 + (-2.80 + 0.868i)T + (60.2 - 41.2i)T^{2} \) |
| 79 | \( 1 + (-0.315 - 0.300i)T + (3.64 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-13.9 + 1.94i)T + (79.8 - 22.7i)T^{2} \) |
| 89 | \( 1 + (-7.08 + 3.52i)T + (53.6 - 71.0i)T^{2} \) |
| 97 | \( 1 + (-9.48 + 1.10i)T + (94.4 - 22.2i)T^{2} \) |
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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
−11.66385698908828823814244514752, −10.65673586883464545500615649108, −9.349057368286512577813105931465, −8.678829829314046697041230204672, −8.549043707241862640961910075736, −6.39187993772848980087760457865, −5.48379657838936210333731679788, −4.78956403024018572932673007645, −1.92963622584403579668148405230, −1.20904667797264073111858755586,
1.64637794460411054794744606995, 3.70695588654262075096580646547, 5.15212354134185788209250563075, 6.40111452433884764859230776831, 7.18193004690132948794287690821, 8.168632077181168553320784630641, 9.258063112808086463055301923232, 10.49971102986668486990438851452, 10.66306859073798284371998882457, 11.39226953432484990830162450431
