| L(s) = 1 | + 3·2-s + 6·4-s + 3·5-s − 3·7-s + 10·8-s + 9·10-s − 3·11-s − 9·14-s + 15·16-s + 8·17-s − 2·19-s + 18·20-s − 9·22-s − 2·23-s + 6·25-s − 18·28-s − 4·29-s + 18·31-s + 21·32-s + 24·34-s − 9·35-s + 4·37-s − 6·38-s + 30·40-s + 18·41-s + 8·43-s − 18·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 1.34·5-s − 1.13·7-s + 3.53·8-s + 2.84·10-s − 0.904·11-s − 2.40·14-s + 15/4·16-s + 1.94·17-s − 0.458·19-s + 4.02·20-s − 1.91·22-s − 0.417·23-s + 6/5·25-s − 3.40·28-s − 0.742·29-s + 3.23·31-s + 3.71·32-s + 4.11·34-s − 1.52·35-s + 0.657·37-s − 0.973·38-s + 4.74·40-s + 2.81·41-s + 1.21·43-s − 2.71·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{3} \cdot 7^{3} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(40.70156992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(40.70156992\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 11 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 13 | $S_4\times C_2$ | \( 1 - T^{2} - 32 T^{3} - p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 8 T + 39 T^{2} - 144 T^{3} + 39 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 27 T^{2} + 20 T^{3} + 27 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 39 T^{2} + 36 T^{3} + 39 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 5 T^{2} - 196 T^{3} + 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 172 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1636 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 8 T + 101 T^{2} - 672 T^{3} + 101 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 93 T^{2} + 252 T^{3} + 93 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 4 T + 77 T^{2} - 4 T^{3} + 77 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 113 T^{2} + 572 T^{3} + 113 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 20 T + 267 T^{2} - 2504 T^{3} + 267 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{3} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 109 T^{2} + 1104 T^{3} + 109 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 4 T + 79 T^{2} - 200 T^{3} + 79 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T - 25 T^{2} - 1324 T^{3} - 25 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 24 T + 401 T^{2} - 4144 T^{3} + 401 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 239 T^{2} - 1028 T^{3} + 239 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 8 T + 281 T^{2} - 1452 T^{3} + 281 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.12304535333699510925728002923, −6.40118342994343550661346774830, −6.30773720406376424542122735294, −6.29810442884946762081809377875, −5.94723572993271944544282425128, −5.88767071404457415589519631135, −5.81619057481646662644419545789, −5.34633783600358587387098007153, −5.02604072181494708509679621393, −4.99152166256193596949802684317, −4.56794340364590913301575095103, −4.50498871607191616874244517904, −4.22397055680046147324046886656, −3.61630609165300829725215484415, −3.59336878002083176972609905955, −3.47968794812256916267234872759, −2.94938794828874688873261728694, −2.83715568147452915297999645983, −2.55865152136873985618501282992, −2.27466995540642455104683326170, −2.04719756610334211928483101543, −1.87112572570495765603572695450, −0.964655340803238414798139798134, −0.846525447533695362917883299068, −0.77097161740939504537771201665,
0.77097161740939504537771201665, 0.846525447533695362917883299068, 0.964655340803238414798139798134, 1.87112572570495765603572695450, 2.04719756610334211928483101543, 2.27466995540642455104683326170, 2.55865152136873985618501282992, 2.83715568147452915297999645983, 2.94938794828874688873261728694, 3.47968794812256916267234872759, 3.59336878002083176972609905955, 3.61630609165300829725215484415, 4.22397055680046147324046886656, 4.50498871607191616874244517904, 4.56794340364590913301575095103, 4.99152166256193596949802684317, 5.02604072181494708509679621393, 5.34633783600358587387098007153, 5.81619057481646662644419545789, 5.88767071404457415589519631135, 5.94723572993271944544282425128, 6.29810442884946762081809377875, 6.30773720406376424542122735294, 6.40118342994343550661346774830, 7.12304535333699510925728002923
