L(s) = 1 | − 2.17·2-s + 2.73·4-s + (0.634 + 1.09i)5-s − 1.60·8-s + (−1.38 − 2.39i)10-s + (−2.73 + 4.74i)11-s + (2.37 − 4.10i)13-s − 1.98·16-s + (−2.40 − 4.17i)17-s + (−2.69 + 4.66i)19-s + (1.73 + 3.00i)20-s + (5.96 − 10.3i)22-s + (−2.58 − 4.48i)23-s + (1.69 − 2.93i)25-s + (−5.16 + 8.94i)26-s + ⋯ |
L(s) = 1 | − 1.53·2-s + 1.36·4-s + (0.283 + 0.491i)5-s − 0.566·8-s + (−0.436 − 0.755i)10-s + (−0.825 + 1.43i)11-s + (0.658 − 1.13i)13-s − 0.496·16-s + (−0.584 − 1.01i)17-s + (−0.617 + 1.06i)19-s + (0.388 + 0.672i)20-s + (1.27 − 2.20i)22-s + (−0.539 − 0.934i)23-s + (0.339 − 0.587i)25-s + (−1.01 + 1.75i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3717603553\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3717603553\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.17T + 2T^{2} \) |
| 5 | \( 1 + (-0.634 - 1.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.73 - 4.74i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.37 + 4.10i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.40 + 4.17i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.58 + 4.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.01 + 3.49i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 1.46T + 31T^{2} \) |
| 37 | \( 1 + (0.959 - 1.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.94 + 3.37i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.66 + 2.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 + (3.57 + 6.18i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 0.308T + 59T^{2} \) |
| 61 | \( 1 - 10.3T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + (-5.27 - 9.13i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9.01T + 79T^{2} \) |
| 83 | \( 1 + (-5.08 - 8.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.59 + 4.49i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (2.48 + 4.30i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.632039493058723460189888322407, −8.448374339133933881090274331384, −8.055398055664410162780954073213, −7.15023164835146517788439547939, −6.51365721676189764294303822167, −5.36877290538012592206552658400, −4.22318589852873368445226603765, −2.66336713272506180517736590661, −1.93194647202227795300748790721, −0.27500788971489958213150706173,
1.19228365503846497788683245207, 2.20202159960256898861694895872, 3.61910561836811450481430982662, 4.87188548064872562313061804492, 5.99571050694130344955325921582, 6.71729925493601148586256022853, 7.74614506656672308584971987991, 8.515837919344236261002968204099, 8.951135250541451724369483562480, 9.545447342787389428446439379469