L(s) = 1 | + 2.38·2-s + 3.69·4-s + (−1.46 − 2.52i)5-s + (−0.138 + 2.64i)7-s + 4.05·8-s + (−3.48 − 6.03i)10-s + (−0.676 + 1.17i)11-s + (−0.733 + 1.26i)13-s + (−0.330 + 6.30i)14-s + 2.27·16-s + (−1.65 − 2.86i)17-s + (−1.10 + 1.91i)19-s + (−5.39 − 9.35i)20-s + (−1.61 + 2.79i)22-s + (1.31 + 2.27i)23-s + ⋯ |
L(s) = 1 | + 1.68·2-s + 1.84·4-s + (−0.653 − 1.13i)5-s + (−0.0523 + 0.998i)7-s + 1.43·8-s + (−1.10 − 1.90i)10-s + (−0.204 + 0.353i)11-s + (−0.203 + 0.352i)13-s + (−0.0883 + 1.68i)14-s + 0.568·16-s + (−0.401 − 0.695i)17-s + (−0.253 + 0.438i)19-s + (−1.20 − 2.09i)20-s + (−0.344 + 0.596i)22-s + (0.274 + 0.474i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.46367 - 0.226450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.46367 - 0.226450i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.138 - 2.64i)T \) |
good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 5 | \( 1 + (1.46 + 2.52i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.676 - 1.17i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.733 - 1.26i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.65 + 2.86i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.10 - 1.91i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.31 - 2.27i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.521 + 0.903i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + (-5.43 + 9.41i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.904 + 1.56i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.17 + 3.76i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.97T + 47T^{2} \) |
| 53 | \( 1 + (-3.22 - 5.59i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 0.559T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 + (-5.22 - 9.05i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 0.767T + 79T^{2} \) |
| 83 | \( 1 + (-0.983 - 1.70i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.20 - 5.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.14 + 7.17i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.56669067070254788198295707452, −11.98129430808715641295444244805, −11.24531742094115863617584778001, −9.481543599625953815089241097231, −8.414258267649646609343281786472, −7.08798778073465736468950056317, −5.73897066403715048473192852532, −4.92054284182461805655849682452, −3.98772343880079845259406960332, −2.40059731473736897453004551597,
2.81593262795084848068713332612, 3.75844561925907912323723201425, 4.79391543438098842522958777063, 6.34392998516726295123224008965, 6.97100462461788598646928278002, 8.108040667549840257939670287581, 10.18931373718056202572058454121, 11.02828419611295588321204321418, 11.62366028669013314807228622738, 12.92860517109563270801519812148