L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + 10-s + (−1.67 − 0.965i)11-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s + 0.517·29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯ |
L(s) = 1 | + (0.965 + 0.258i)2-s + (0.866 + 0.499i)4-s + (0.965 − 0.258i)5-s + (0.5 − 0.866i)7-s + (0.707 + 0.707i)8-s + 10-s + (−1.67 − 0.965i)11-s + (0.707 − 0.707i)14-s + (0.500 + 0.866i)16-s + (0.965 + 0.258i)20-s + (−1.36 − 1.36i)22-s + (0.866 − 0.499i)25-s + (0.866 − 0.5i)28-s + 0.517·29-s + (−0.866 + 1.5i)31-s + (0.258 + 0.965i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0333i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.551488575\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.551488575\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.965 + 0.258i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 11 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 - 0.517T + T^{2} \) |
| 31 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (1.67 - 0.448i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-1.36 - 1.36i)T + iT^{2} \) |
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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
−8.896779847237844755552111180080, −8.140328213295771209651050550866, −7.50756306209530348017447183920, −6.64145251167816093723430104378, −5.80278235551575740558083633432, −5.17022351319564278926726290428, −4.59051324897580181205242982269, −3.37168827378705085411580815639, −2.60731449690785535209206245488, −1.44081591806702413181888202336,
1.87353258294141004630001779394, 2.34034423307549558354033516478, 3.20731082282825408458241050309, 4.63418974721734456911770805032, 5.14054981906372861237618494669, 5.79303016096554265396449072370, 6.52844323394403133100358310198, 7.51378436384423260718728924490, 8.133270440955267772559678164863, 9.457189813153641840832465457328