L(s) = 1 | + (0.707 + 0.707i)2-s + (1.47 + 1.47i)3-s + 1.00i·4-s + (−2.23 − 0.0355i)5-s + 2.08i·6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + 1.35i·9-s + (−1.55 − 1.60i)10-s + (0.318 + 3.30i)11-s + (−1.47 + 1.47i)12-s + (−2.95 + 2.95i)13-s + 1.00i·14-s + (−3.24 − 3.35i)15-s − 1.00·16-s + (−0.451 − 0.451i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.852 + 0.852i)3-s + 0.500i·4-s + (−0.999 − 0.0158i)5-s + 0.852i·6-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + 0.451i·9-s + (−0.491 − 0.507i)10-s + (0.0959 + 0.995i)11-s + (−0.426 + 0.426i)12-s + (−0.819 + 0.819i)13-s + 0.267i·14-s + (−0.838 − 0.865i)15-s − 0.250·16-s + (−0.109 − 0.109i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.457591 + 1.89726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.457591 + 1.89726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (2.23 + 0.0355i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (-0.318 - 3.30i)T \) |
good | 3 | \( 1 + (-1.47 - 1.47i)T + 3iT^{2} \) |
| 13 | \( 1 + (2.95 - 2.95i)T - 13iT^{2} \) |
| 17 | \( 1 + (0.451 + 0.451i)T + 17iT^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 + (-1.62 - 1.62i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.38T + 29T^{2} \) |
| 31 | \( 1 - 3.62T + 31T^{2} \) |
| 37 | \( 1 + (0.785 - 0.785i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.52iT - 41T^{2} \) |
| 43 | \( 1 + (0.646 - 0.646i)T - 43iT^{2} \) |
| 47 | \( 1 + (0.355 - 0.355i)T - 47iT^{2} \) |
| 53 | \( 1 + (-7.85 - 7.85i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.53iT - 59T^{2} \) |
| 61 | \( 1 + 3.09iT - 61T^{2} \) |
| 67 | \( 1 + (-11.1 + 11.1i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + (-1.42 + 1.42i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.62T + 79T^{2} \) |
| 83 | \( 1 + (-5.82 + 5.82i)T - 83iT^{2} \) |
| 89 | \( 1 + 2.20iT - 89T^{2} \) |
| 97 | \( 1 + (-10.4 + 10.4i)T - 97iT^{2} \) |
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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
−10.59365824128803018063413396501, −9.612437500687442500583912780350, −8.895953530636932443417892659561, −8.134138327046004859489259431949, −7.29246148362854216789602776873, −6.44647089932378453464855231838, −4.72100382376445567816793151276, −4.53287043819842012351023182321, −3.48133501564525295692563251006, −2.36721250641244207044028971626,
0.75924489140395243744331996022, 2.35533595399358033080667290705, 3.21235414470991116074653597543, 4.22495468367510825440990384964, 5.31205234347408742823117695872, 6.64729372521047278200530056241, 7.43026318519134942103002588995, 8.328442394269862044308961407951, 8.719169596465681853361719581492, 10.22431230325689533358787838676