L(s) = 1 | − i·3-s − 9-s − 2i·13-s − 6i·17-s + 4·19-s − 8i·23-s + i·27-s − 2·29-s − 4·31-s + 10i·37-s − 2·39-s + 2·41-s − 4i·43-s + 8i·47-s + 7·49-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.333·9-s − 0.554i·13-s − 1.45i·17-s + 0.917·19-s − 1.66i·23-s + 0.192i·27-s − 0.371·29-s − 0.718·31-s + 1.64i·37-s − 0.320·39-s + 0.312·41-s − 0.609i·43-s + 1.16i·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.200021513\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.200021513\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−7.78836093625334776465573551175, −7.34548609825308846561796589929, −6.60784960757236337997500695256, −5.82004934949027835210529867676, −5.08651605796836415823035017968, −4.33034270936793254315718176937, −3.09088517238200020554199613429, −2.62711201163926188229327463379, −1.36090735493308643597599855026, −0.33542539082138709577253890089,
1.36201600088555421932110646520, 2.31624474995359990651049566964, 3.66291170833758050763207099438, 3.79719581801783884373091038081, 4.97780553932737593400022108817, 5.63527264948320825461346414328, 6.25449061157180928154606929417, 7.34554876111416156679978408542, 7.73883503057465455724837766867, 8.810693558670117024080883748727