L(s) = 1 | + (−0.800 − 0.800i)2-s + (1.34 + 1.09i)3-s − 0.718i·4-s + (2.10 − 0.754i)5-s + (−0.199 − 1.95i)6-s + (−2.17 + 2.17i)8-s + (0.606 + 2.93i)9-s + (−2.28 − 1.08i)10-s − 5.20i·11-s + (0.785 − 0.964i)12-s + (3.24 + 3.24i)13-s + (3.65 + 1.28i)15-s + 2.04·16-s + (0.844 + 0.844i)17-s + (1.86 − 2.83i)18-s − 1.32i·19-s + ⋯ |
L(s) = 1 | + (−0.566 − 0.566i)2-s + (0.775 + 0.631i)3-s − 0.359i·4-s + (0.941 − 0.337i)5-s + (−0.0813 − 0.796i)6-s + (−0.769 + 0.769i)8-s + (0.202 + 0.979i)9-s + (−0.723 − 0.341i)10-s − 1.56i·11-s + (0.226 − 0.278i)12-s + (0.900 + 0.900i)13-s + (0.942 + 0.332i)15-s + 0.511·16-s + (0.204 + 0.204i)17-s + (0.439 − 0.668i)18-s − 0.302i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.57724 - 0.754951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.57724 - 0.754951i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 - 1.09i)T \) |
| 5 | \( 1 + (-2.10 + 0.754i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.800 + 0.800i)T + 2iT^{2} \) |
| 11 | \( 1 + 5.20iT - 11T^{2} \) |
| 13 | \( 1 + (-3.24 - 3.24i)T + 13iT^{2} \) |
| 17 | \( 1 + (-0.844 - 0.844i)T + 17iT^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 + (-5.62 + 5.62i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.38T + 29T^{2} \) |
| 31 | \( 1 - 1.70T + 31T^{2} \) |
| 37 | \( 1 + (1.71 - 1.71i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 + (0.281 + 0.281i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.39 + 3.39i)T + 47iT^{2} \) |
| 53 | \( 1 + (3.51 - 3.51i)T - 53iT^{2} \) |
| 59 | \( 1 - 1.81T + 59T^{2} \) |
| 61 | \( 1 - 2.47T + 61T^{2} \) |
| 67 | \( 1 + (-7.92 + 7.92i)T - 67iT^{2} \) |
| 71 | \( 1 - 9.06iT - 71T^{2} \) |
| 73 | \( 1 + (-1.33 - 1.33i)T + 73iT^{2} \) |
| 79 | \( 1 + 11.5iT - 79T^{2} \) |
| 83 | \( 1 + (5.46 - 5.46i)T - 83iT^{2} \) |
| 89 | \( 1 + 9.43T + 89T^{2} \) |
| 97 | \( 1 + (-3.06 + 3.06i)T - 97iT^{2} \) |
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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
−10.18619781646947964593729734523, −9.338405529425649503968927933661, −8.768170334170019631429187422050, −8.355341869776696926655404441530, −6.59603928117687225803233567265, −5.73930538679153337674186950927, −4.80794397420283644588761335298, −3.41346697885952917460540685166, −2.38783585741487481787517012213, −1.18847020337982519990398855579,
1.48984179150429415229118628729, 2.78101436619476388516100391720, 3.72391408152466058644818580549, 5.41081020811272355888957489441, 6.48597971523380984950938821457, 7.19171787973155928870399709727, 7.78955468309050485786700393877, 8.753585860679695971085446817826, 9.504573615934410528771131414071, 10.01627486977073900103401158069