L(s) = 1 | + 0.239·2-s + (−1.70 + 0.315i)3-s − 1.94·4-s + (−1.29 + 2.24i)5-s + (−0.407 + 0.0753i)6-s − 0.942·8-s + (2.80 − 1.07i)9-s + (−0.309 + 0.536i)10-s + (−2.09 − 3.62i)11-s + (3.30 − 0.612i)12-s + (1.84 + 3.18i)13-s + (1.5 − 4.23i)15-s + 3.66·16-s + (0.855 − 1.48i)17-s + (0.669 − 0.256i)18-s + (−3.57 − 6.19i)19-s + ⋯ |
L(s) = 1 | + 0.169·2-s + (−0.983 + 0.181i)3-s − 0.971·4-s + (−0.579 + 1.00i)5-s + (−0.166 + 0.0307i)6-s − 0.333·8-s + (0.933 − 0.357i)9-s + (−0.0979 + 0.169i)10-s + (−0.630 − 1.09i)11-s + (0.955 − 0.176i)12-s + (0.510 + 0.884i)13-s + (0.387 − 1.09i)15-s + 0.915·16-s + (0.207 − 0.359i)17-s + (0.157 − 0.0604i)18-s + (−0.820 − 1.42i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.373018 - 0.269913i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.373018 - 0.269913i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.70 - 0.315i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 + (1.29 - 2.24i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (2.09 + 3.62i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.84 - 3.18i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.855 + 1.48i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.57 + 6.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.56 + 4.43i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.06 + 1.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 + (0.830 + 1.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.10 + 8.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.830 + 1.43i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 9.33T + 47T^{2} \) |
| 53 | \( 1 + (5.32 - 9.22i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + (-3.57 + 6.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + (-3.44 + 5.97i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.51 + 4.36i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.53 + 2.65i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.98965510882528535880732257740, −10.33585905504716791263211548079, −9.155196764271623778914442698480, −8.281258575032012452641018762248, −6.97205455280564131551076371233, −6.24631630563273861014735150891, −5.06428997967944020175069478968, −4.19616083194536822458059389032, −3.05517595076432151542416738989, −0.37220040565891444833985539248,
1.24737847129963641946849912318, 3.72560182670650211030794877481, 4.77306242370155286019633368997, 5.26846586961445091347403984948, 6.42562967403309082191236267146, 7.988423566422707633172978702132, 8.241632252421498908838433170291, 9.753691305685788532744587156434, 10.25518066981165553977525519739, 11.47738900162174380659430182188