| L(s) = 1 | − 4·5-s + 2·11-s + 4·13-s − 6·17-s + 4·19-s − 4·23-s + 4·25-s − 3·29-s + 14·31-s − 2·37-s − 10·41-s + 6·43-s − 2·47-s − 21·49-s − 8·55-s + 8·59-s + 2·61-s − 16·65-s − 20·67-s + 12·71-s + 2·73-s + 30·79-s + 32·83-s + 24·85-s − 10·89-s − 16·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s + 1.10·13-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 4/5·25-s − 0.557·29-s + 2.51·31-s − 0.328·37-s − 1.56·41-s + 0.914·43-s − 0.291·47-s − 3·49-s − 1.07·55-s + 1.04·59-s + 0.256·61-s − 1.98·65-s − 2.44·67-s + 1.42·71-s + 0.234·73-s + 3.37·79-s + 3.51·83-s + 2.60·85-s − 1.05·89-s − 1.64·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.248078252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.248078252\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 4 T + 12 T^{2} + 6 p T^{3} + 12 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 - 2 T + 4 T^{2} + 36 T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 T + 20 T^{2} - 102 T^{3} + 20 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 - 4 T + 29 T^{2} - 120 T^{3} + 29 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 120 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 14 T + 152 T^{2} - 936 T^{3} + 152 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2 T + 79 T^{2} + 116 T^{3} + 79 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 59 T^{2} + 308 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 6 T + 92 T^{2} - 548 T^{3} + 92 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 24 T^{2} - 264 T^{3} + 24 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 116 T^{2} - 58 T^{3} + 116 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 173 T^{2} - 928 T^{3} + 173 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 2 T + 83 T^{2} + 84 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 20 T + 233 T^{2} + 2040 T^{3} + 233 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 12 T + 65 T^{2} - 8 T^{3} + 65 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 2 T + 123 T^{2} - 452 T^{3} + 123 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 30 T + 488 T^{2} - 5128 T^{3} + 488 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 32 T + 565 T^{2} - 6288 T^{3} + 565 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 10 T + 11 T^{2} - 1036 T^{3} + 11 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 323 T^{2} + 2652 T^{3} + 323 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.60042949801138548604385102322, −7.07718311462320530317035542398, −6.94376033076710798673056167088, −6.71478920997882914267176387218, −6.44960504187809977407494200763, −6.42608391395750011003931778079, −6.02306507593483420250127305832, −5.73574380645117077003005227651, −5.47799223374428180968531762040, −5.00768151462175648179643548519, −4.82192809418148160573853727468, −4.55375571780562476794899598673, −4.48935858807372632311785772787, −3.90698566962958047533184160826, −3.79119697990933264293744921432, −3.77076401327842547761460117824, −3.21437331220507161843547041853, −3.06309337660594144777521636544, −2.88345889500442868621409526251, −2.11302334457760663719050282111, −1.89465183393304766554792729360, −1.80163102510392151767884283348, −0.959760639146296396183830089581, −0.830705945922961809383246498234, −0.26621199965713702804524663444,
0.26621199965713702804524663444, 0.830705945922961809383246498234, 0.959760639146296396183830089581, 1.80163102510392151767884283348, 1.89465183393304766554792729360, 2.11302334457760663719050282111, 2.88345889500442868621409526251, 3.06309337660594144777521636544, 3.21437331220507161843547041853, 3.77076401327842547761460117824, 3.79119697990933264293744921432, 3.90698566962958047533184160826, 4.48935858807372632311785772787, 4.55375571780562476794899598673, 4.82192809418148160573853727468, 5.00768151462175648179643548519, 5.47799223374428180968531762040, 5.73574380645117077003005227651, 6.02306507593483420250127305832, 6.42608391395750011003931778079, 6.44960504187809977407494200763, 6.71478920997882914267176387218, 6.94376033076710798673056167088, 7.07718311462320530317035542398, 7.60042949801138548604385102322
