L(s) = 1 | + 2-s + 4-s + (−0.794 − 1.37i)5-s + 8-s + (−0.794 − 1.37i)10-s + (−0.794 + 1.37i)11-s + (−2.40 + 4.16i)13-s + 16-s + (−2.69 − 4.67i)17-s + (3.54 − 6.14i)19-s + (−0.794 − 1.37i)20-s + (−0.794 + 1.37i)22-s + (0.150 + 0.260i)23-s + (1.23 − 2.14i)25-s + (−2.40 + 4.16i)26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.355 − 0.615i)5-s + 0.353·8-s + (−0.251 − 0.434i)10-s + (−0.239 + 0.414i)11-s + (−0.667 + 1.15i)13-s + 0.250·16-s + (−0.654 − 1.13i)17-s + (0.814 − 1.41i)19-s + (−0.177 − 0.307i)20-s + (−0.169 + 0.293i)22-s + (0.0313 + 0.0542i)23-s + (0.247 − 0.429i)25-s + (−0.471 + 0.817i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.959310781\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.959310781\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.794 + 1.37i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.794 - 1.37i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.40 - 4.16i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.69 + 4.67i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.54 + 6.14i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.150 - 0.260i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.13 + 7.16i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2.71T + 31T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.93 + 5.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.833 + 1.44i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 + (2.44 + 4.23i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.47T + 59T^{2} \) |
| 61 | \( 1 - 4.47T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 + (8.02 + 13.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 8.38T + 79T^{2} \) |
| 83 | \( 1 + (-1.18 - 2.04i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.60 + 2.78i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.712 + 1.23i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.793767359598425702965963708698, −7.57733240086590995935531293260, −7.18668953712356965599026889407, −6.36527757898199547279125141536, −5.25098212574126370373068025264, −4.66081946104972474058669399303, −4.15035304989601197833022837080, −2.82907377672519566517397395772, −2.08489891374438329445174584734, −0.48954157399292824475978867295,
1.42820842158231725149751660901, 2.77411492085323730793495917982, 3.36658373412210745385703383236, 4.20593193592584605247715076513, 5.31109376289310881764007882691, 5.82736829883518280664212743225, 6.72322618717048950595320867380, 7.56600303669821737611223798748, 8.025875713958985525182077890056, 8.988625942720993642611434570366