L(s) = 1 | + (0.232 + 1.71i)3-s + (−0.581 − 0.335i)5-s + (2.63 − 0.209i)7-s + (−2.89 + 0.798i)9-s + (2.62 + 4.54i)11-s + 2·13-s + (0.440 − 1.07i)15-s + (−3.64 + 2.10i)17-s + (1.13 + 0.656i)19-s + (0.972 + 4.47i)21-s + (−1.60 + 2.77i)23-s + (−2.27 − 3.94i)25-s + (−2.04 − 4.77i)27-s + 6.22i·29-s + (1.5 − 0.866i)31-s + ⋯ |
L(s) = 1 | + (0.134 + 0.990i)3-s + (−0.259 − 0.150i)5-s + (0.996 − 0.0791i)7-s + (−0.963 + 0.266i)9-s + (0.791 + 1.37i)11-s + 0.554·13-s + (0.113 − 0.277i)15-s + (−0.884 + 0.510i)17-s + (0.260 + 0.150i)19-s + (0.212 + 0.977i)21-s + (−0.334 + 0.578i)23-s + (−0.454 − 0.788i)25-s + (−0.393 − 0.919i)27-s + 1.15i·29-s + (0.269 − 0.155i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.12154 + 0.874473i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.12154 + 0.874473i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.232 - 1.71i)T \) |
| 7 | \( 1 + (-2.63 + 0.209i)T \) |
good | 5 | \( 1 + (0.581 + 0.335i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.62 - 4.54i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (3.64 - 2.10i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.13 - 0.656i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.60 - 2.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.22iT - 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.13 + 5.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 9.76iT - 41T^{2} \) |
| 43 | \( 1 - 9.55iT - 43T^{2} \) |
| 47 | \( 1 + (-4.80 + 8.32i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.1 + 5.88i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (3.78 + 6.55i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.86 + 3.22i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.58 - 1.49i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.08T + 71T^{2} \) |
| 73 | \( 1 + (0.137 + 0.238i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 2.04T + 83T^{2} \) |
| 89 | \( 1 + (12.1 + 6.98i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 3.27T + 97T^{2} \) |
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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
−11.59000793524087541394340576564, −10.83091068284289721568920205866, −9.911211836463938227172915840956, −8.967090521573165096776396237259, −8.200695010204104380928507120724, −7.04678970499621186189146857368, −5.63390482858984553750201286481, −4.49567771544179467334664980088, −3.89191362505055051542278292979, −1.98232555184442096963664477382,
1.12962487796893792540139235923, 2.70498399085605351084742013628, 4.14318501168961006871465519867, 5.68802885279970296774255418691, 6.52901101556257307377067880454, 7.65397794386369086107476923757, 8.439634118564881917328832867656, 9.136746211206989543087930308619, 10.83308325183741600356503958043, 11.53473872440541905373630938552