| L(s) = 1 | − 6·11-s + 6·17-s − 4·19-s − 6·23-s + 2·29-s + 8·31-s + 2·37-s − 10·41-s − 12·43-s − 8·47-s − 2·53-s + 4·59-s − 8·61-s − 16·67-s − 10·71-s − 4·79-s + 4·83-s + 6·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 0.371·29-s + 1.43·31-s + 0.328·37-s − 1.56·41-s − 1.82·43-s − 1.16·47-s − 0.274·53-s + 0.520·59-s − 1.02·61-s − 1.95·67-s − 1.18·71-s − 0.450·79-s + 0.439·83-s + 0.635·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3733844786\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3733844786\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.24555817483702, −12.70998992907502, −12.26262370151425, −11.70875636393667, −11.51476283230730, −10.46792524696780, −10.33227277598635, −10.12196511197538, −9.540469369709289, −8.682367217603270, −8.297441345053077, −7.936370167681732, −7.588202217365496, −6.869542681550973, −6.247616465981845, −5.915804506342297, −5.248946233217378, −4.799151522223087, −4.407716414723553, −3.439729520541052, −3.153904853675878, −2.523622612678140, −1.870552490168727, −1.249195871443723, −0.1733862371478516,
0.1733862371478516, 1.249195871443723, 1.870552490168727, 2.523622612678140, 3.153904853675878, 3.439729520541052, 4.407716414723553, 4.799151522223087, 5.248946233217378, 5.915804506342297, 6.247616465981845, 6.869542681550973, 7.588202217365496, 7.936370167681732, 8.297441345053077, 8.682367217603270, 9.540469369709289, 10.12196511197538, 10.33227277598635, 10.46792524696780, 11.51476283230730, 11.70875636393667, 12.26262370151425, 12.70998992907502, 13.24555817483702
