L(s) = 1 | + (−0.707 + 0.707i)5-s + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (1 − i)37-s + 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s + 1.41·65-s + (−1 − i)73-s + 2.00·85-s − 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)5-s + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.00i·25-s − 1.41·29-s + (1 − i)37-s + 1.41i·41-s − i·49-s + (1.41 − 1.41i)53-s + 1.41·65-s + (−1 − i)73-s + 2.00·85-s − 1.41·89-s + (−1 + i)97-s − 1.41i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4821272437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4821272437\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.707 - 0.707i)T \) |
good | 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (1 + i)T + iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-1 + i)T - iT^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + (1 - i)T - iT^{2} \) |
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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
−8.673406145824898526798083348495, −7.78889848193286770418377033849, −7.26948953256666789895278007416, −6.66283427001140484210764452024, −5.59754026359606987233209693431, −4.78893589283817434425890294574, −3.95075396411524422332402542777, −2.93447964331481709106596383525, −2.28726575829011616403461018623, −0.29006347170982799871744771452,
1.55348687072724924556698143088, 2.55133007551641436562008994529, 4.04839360004186556122114587147, 4.23527474659811178759394437644, 5.25573181463499213637596401134, 6.15756211351203787298141746893, 7.06964687182244475876745219671, 7.64418163724265159091032604958, 8.587333095900638290548022082971, 9.011949955938875133116711771782