L(s) = 1 | + (−1.11 − 1.11i)2-s + (−1.51 − 0.843i)3-s + 0.487i·4-s + (0.746 + 2.62i)6-s + (0.707 − 0.707i)7-s + (−1.68 + 1.68i)8-s + (1.57 + 2.55i)9-s + 2.87i·11-s + (0.411 − 0.737i)12-s + (−1.36 − 1.36i)13-s − 1.57·14-s + 4.73·16-s + (4.87 + 4.87i)17-s + (1.08 − 4.60i)18-s + 3.03i·19-s + ⋯ |
L(s) = 1 | + (−0.788 − 0.788i)2-s + (−0.873 − 0.486i)3-s + 0.243i·4-s + (0.304 + 1.07i)6-s + (0.267 − 0.267i)7-s + (−0.596 + 0.596i)8-s + (0.525 + 0.850i)9-s + 0.865i·11-s + (0.118 − 0.212i)12-s + (−0.378 − 0.378i)13-s − 0.421·14-s + 1.18·16-s + (1.18 + 1.18i)17-s + (0.256 − 1.08i)18-s + 0.697i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.583687 - 0.111486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.583687 - 0.111486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.51 + 0.843i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.11 + 1.11i)T + 2iT^{2} \) |
| 11 | \( 1 - 2.87iT - 11T^{2} \) |
| 13 | \( 1 + (1.36 + 1.36i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.87 - 4.87i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.03iT - 19T^{2} \) |
| 23 | \( 1 + (4.40 - 4.40i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + (-4.41 + 4.41i)T - 37iT^{2} \) |
| 41 | \( 1 + 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (-4.59 - 4.59i)T + 43iT^{2} \) |
| 47 | \( 1 + (-6.91 - 6.91i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.23 + 2.23i)T - 53iT^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + (11.0 - 11.0i)T - 67iT^{2} \) |
| 71 | \( 1 + 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (-4.85 - 4.85i)T + 73iT^{2} \) |
| 79 | \( 1 - 11.3iT - 79T^{2} \) |
| 83 | \( 1 + (-7.16 + 7.16i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.776T + 89T^{2} \) |
| 97 | \( 1 + (-1.56 + 1.56i)T - 97iT^{2} \) |
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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
−10.56161621958328467689345182139, −10.25806994697432023909459914019, −9.365831727314376890838368135918, −7.968897063232124211165589269868, −7.52669537236619524379408720262, −6.03835776699167377893349790644, −5.41716306164668908130061724387, −3.97827916897096262209070682731, −2.19099548346945399073393209007, −1.17700461882032889629905182432,
0.60360017085157915260320554262, 3.07203682875451791139303789092, 4.46368552707945434303100858978, 5.58325785502000784990801466007, 6.37009208823118979308613001839, 7.30109543597743579416117076158, 8.214818716040126492888311586776, 9.196534166517895895575296795500, 9.799481885884983083294947138799, 10.77379686827809229780275232848