L(s) = 1 | + 1.86i·2-s + (1.72 + 0.174i)3-s − 1.49·4-s + (−1.25 − 2.17i)5-s + (−0.326 + 3.22i)6-s + 0.947i·8-s + (2.93 + 0.601i)9-s + (4.05 − 2.34i)10-s + (4.85 + 2.80i)11-s + (−2.57 − 0.260i)12-s + (−0.384 − 0.221i)13-s + (−1.78 − 3.95i)15-s − 4.75·16-s + (1.53 + 2.66i)17-s + (−1.12 + 5.49i)18-s + (2.22 + 1.28i)19-s + ⋯ |
L(s) = 1 | + 1.32i·2-s + (0.994 + 0.100i)3-s − 0.746·4-s + (−0.560 − 0.970i)5-s + (−0.133 + 1.31i)6-s + 0.335i·8-s + (0.979 + 0.200i)9-s + (1.28 − 0.740i)10-s + (1.46 + 0.845i)11-s + (−0.742 − 0.0751i)12-s + (−0.106 − 0.0615i)13-s + (−0.459 − 1.02i)15-s − 1.18·16-s + (0.373 + 0.646i)17-s + (−0.264 + 1.29i)18-s + (0.511 + 0.295i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26423 + 1.45661i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26423 + 1.45661i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.72 - 0.174i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 1.86iT - 2T^{2} \) |
| 5 | \( 1 + (1.25 + 2.17i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.85 - 2.80i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.384 + 0.221i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.53 - 2.66i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.22 - 1.28i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (6.83 - 3.94i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.71 + 1.56i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.64 - 2.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.75 + 8.23i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 + (-4.20 + 2.42i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 7.30T + 59T^{2} \) |
| 61 | \( 1 + 8.55iT - 61T^{2} \) |
| 67 | \( 1 + 1.86T + 67T^{2} \) |
| 71 | \( 1 - 2.95iT - 71T^{2} \) |
| 73 | \( 1 + (7.37 - 4.25i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 0.574T + 79T^{2} \) |
| 83 | \( 1 + (4.23 + 7.33i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.78 - 6.56i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.22 + 1.86i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.67440449940674884103350678516, −9.928885329819561549807110677764, −9.275733685088843034714372528553, −8.321112260829252908758998580147, −7.84718730665963856154193050142, −6.94243952216263650250860703759, −5.83411533229806658086434449960, −4.53239649015311366868373886972, −3.85677872047998405200763087403, −1.80559318869493659802777653488,
1.36608007326560906756154640737, 2.88549496415769422582627603290, 3.39492880997980871258763134369, 4.37129407201147629212680142187, 6.50436819516078322820997034098, 7.17452332044060973734563736289, 8.417105482924971878822333927517, 9.259232291920838906688633538749, 10.12377821535296417750699897688, 10.88425007297867593226190841671