| L(s) = 1 | + 4·11-s + 2·13-s − 6·17-s − 4·19-s + 4·23-s − 11·25-s + 4·31-s + 10·37-s + 14·41-s − 20·43-s − 16·47-s − 5·49-s + 6·53-s + 24·59-s − 2·61-s + 10·67-s + 12·71-s − 10·73-s − 2·79-s + 4·83-s + 12·89-s + 18·97-s − 20·101-s + 6·103-s + 8·107-s − 6·109-s + 20·113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.917·19-s + 0.834·23-s − 2.19·25-s + 0.718·31-s + 1.64·37-s + 2.18·41-s − 3.04·43-s − 2.33·47-s − 5/7·49-s + 0.824·53-s + 3.12·59-s − 0.256·61-s + 1.22·67-s + 1.42·71-s − 1.17·73-s − 0.225·79-s + 0.439·83-s + 1.27·89-s + 1.82·97-s − 1.99·101-s + 0.591·103-s + 0.773·107-s − 0.574·109-s + 1.88·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 167^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.644573899\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644573899\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 167 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 11 T^{2} + 2 T^{3} + 11 p T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T^{2} + 16 T^{3} + 5 p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 3 p T^{2} - 84 T^{3} + 3 p^{2} T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $D_{6}$ | \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 59 T^{2} + 206 T^{3} + 59 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 4 T + 41 T^{2} + 120 T^{3} + 41 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 21 T^{2} - 120 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 23 T^{2} + 128 T^{3} + 23 p T^{4} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 280 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 10 T + 91 T^{2} - 748 T^{3} + 91 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 14 T + 151 T^{2} - 1026 T^{3} + 151 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 20 T + 241 T^{2} + 1838 T^{3} + 241 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 16 T + 221 T^{2} + 1628 T^{3} + 221 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 59 T^{2} - 170 T^{3} + 59 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 24 T + 353 T^{2} - 3232 T^{3} + 353 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 179 T^{2} + 240 T^{3} + 179 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 10 T + 221 T^{2} - 1314 T^{3} + 221 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 10 T + 231 T^{2} + 1420 T^{3} + 231 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2 T + 37 T^{2} + 650 T^{3} + 37 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 4 T + 161 T^{2} - 680 T^{3} + 161 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 12 T + 251 T^{2} - 1816 T^{3} + 251 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 18 T + 255 T^{2} - 2412 T^{3} + 255 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.14776912905252219079173761069, −6.82169452793044580027582647314, −6.51973096034321351711711539630, −6.50889013627363891060740454717, −6.11124884586916891395447254143, −6.07209263457006207690367153371, −6.03674915500092210776793334278, −5.25107627088333417742413073180, −5.18138049440735798960384844549, −5.05889189731661475147995574307, −4.63653767333569513845908612847, −4.44426471252399524938296933808, −4.11201258441599326333491277896, −3.77354136993346645666045609266, −3.75986858880116846713587247271, −3.64511714781206125420280175629, −2.92676938673234464202461887227, −2.86636030428331692893615148275, −2.44188891824670624370552184764, −2.00276263135305488825405961863, −1.96938113176195731561375231527, −1.64181112840925146420453473338, −0.981666365539030035682791486911, −0.892596863415654562334347988198, −0.22611135603423046277344123224,
0.22611135603423046277344123224, 0.892596863415654562334347988198, 0.981666365539030035682791486911, 1.64181112840925146420453473338, 1.96938113176195731561375231527, 2.00276263135305488825405961863, 2.44188891824670624370552184764, 2.86636030428331692893615148275, 2.92676938673234464202461887227, 3.64511714781206125420280175629, 3.75986858880116846713587247271, 3.77354136993346645666045609266, 4.11201258441599326333491277896, 4.44426471252399524938296933808, 4.63653767333569513845908612847, 5.05889189731661475147995574307, 5.18138049440735798960384844549, 5.25107627088333417742413073180, 6.03674915500092210776793334278, 6.07209263457006207690367153371, 6.11124884586916891395447254143, 6.50889013627363891060740454717, 6.51973096034321351711711539630, 6.82169452793044580027582647314, 7.14776912905252219079173761069
