| 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
−8.810693558670117024080883748727, −7.73883503057465455724837766867, −7.34554876111416156679978408542, −6.25449061157180928154606929417, −5.63527264948320825461346414328, −4.97780553932737593400022108817, −3.79719581801783884373091038081, −3.66291170833758050763207099438, −2.31624474995359990651049566964, −1.36201600088555421932110646520,
0.33542539082138709577253890089, 1.36090735493308643597599855026, 2.62711201163926188229327463379, 3.09088517238200020554199613429, 4.33034270936793254315718176937, 5.08651605796836415823035017968, 5.82004934949027835210529867676, 6.60784960757236337997500695256, 7.34548609825308846561796589929, 7.78836093625334776465573551175
