| L(s) = 1 | + 3.46·5-s − 2i·7-s − 13.8i·11-s + 20.7·13-s + 18·17-s − 20.7i·19-s − 36i·23-s − 13.0·25-s − 31.1·29-s + 22i·31-s − 6.92i·35-s + 41.5·37-s − 54·41-s − 20.7i·43-s + 36i·47-s + ⋯ |
| L(s) = 1 | + 0.692·5-s − 0.285i·7-s − 1.25i·11-s + 1.59·13-s + 1.05·17-s − 1.09i·19-s − 1.56i·23-s − 0.520·25-s − 1.07·29-s + 0.709i·31-s − 0.197i·35-s + 1.12·37-s − 1.31·41-s − 0.483i·43-s + 0.765i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.366808537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366808537\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 3.46T + 25T^{2} \) |
| 7 | \( 1 + 2iT - 49T^{2} \) |
| 11 | \( 1 + 13.8iT - 121T^{2} \) |
| 13 | \( 1 - 20.7T + 169T^{2} \) |
| 17 | \( 1 - 18T + 289T^{2} \) |
| 19 | \( 1 + 20.7iT - 361T^{2} \) |
| 23 | \( 1 + 36iT - 529T^{2} \) |
| 29 | \( 1 + 31.1T + 841T^{2} \) |
| 31 | \( 1 - 22iT - 961T^{2} \) |
| 37 | \( 1 - 41.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 54T + 1.68e3T^{2} \) |
| 43 | \( 1 + 20.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 36iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 100.T + 2.80e3T^{2} \) |
| 59 | \( 1 - 62.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 - 62.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 108iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 + 50iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 13.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 18T + 7.92e3T^{2} \) |
| 97 | \( 1 + 34T + 9.40e3T^{2} \) |
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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
−8.656144221013672993266615795587, −8.032715515381687373144102861733, −6.98632762006000726451559337022, −6.07045181940258711677271354676, −5.78075041926975994405866492294, −4.63502026540535267666422667269, −3.60030202344744116065612202545, −2.87196078648473092238224030275, −1.54133065481607711135447190067, −0.58256539221605729746424592267,
1.39009292994680560335532760675, 1.96081875340444630844393176502, 3.37499114718990551931288490770, 4.03179492751908042416226767094, 5.32972786236860816435231890390, 5.79808770915859475070445701763, 6.55572584399130593812761562655, 7.66314706119941980160393970556, 8.082672844485453059633777211836, 9.261512900199593365952068636083
