| L(s) = 1 | − 2·3-s − 7·7-s − 9-s − 3·13-s + 4·17-s + 8·19-s + 14·21-s − 9·23-s + 7·27-s − 9·29-s + 17·31-s + 3·37-s + 6·39-s − 16·41-s + 4·43-s − 11·47-s + 18·49-s − 8·51-s + 3·53-s − 16·57-s − 2·59-s + 15·61-s + 7·63-s + 5·67-s + 18·69-s − 5·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 2.64·7-s − 1/3·9-s − 0.832·13-s + 0.970·17-s + 1.83·19-s + 3.05·21-s − 1.87·23-s + 1.34·27-s − 1.67·29-s + 3.05·31-s + 0.493·37-s + 0.960·39-s − 2.49·41-s + 0.609·43-s − 1.60·47-s + 18/7·49-s − 1.12·51-s + 0.412·53-s − 2.11·57-s − 0.260·59-s + 1.92·61-s + 0.881·63-s + 0.610·67-s + 2.16·69-s − 0.593·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165685856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165685856\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 2 T + 5 T^{2} + 5 T^{3} + 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + p T + 31 T^{2} + 94 T^{3} + 31 p T^{4} + p^{3} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T^{2} + 27 T^{3} - 3 p T^{4} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 16 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 4 T + 31 T^{2} - 120 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 8 T + 53 T^{2} - 240 T^{3} + 53 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 4 p T^{2} + 428 T^{3} + 4 p^{2} T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 9 T + 110 T^{2} + 536 T^{3} + 110 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 17 T + 184 T^{2} - 1202 T^{3} + 184 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 16 T + 193 T^{2} + 1359 T^{3} + 193 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 4 T + 9 T^{2} - 112 T^{3} + 9 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 11 T + 131 T^{2} + 1030 T^{3} + 131 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 59 T^{2} - 610 T^{3} + 59 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 53 T^{2} + 220 T^{3} + 53 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 15 T + 212 T^{2} - 1778 T^{3} + 212 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 5 T + 22 T^{2} + 274 T^{3} + 22 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 5 T + 189 T^{2} + 714 T^{3} + 189 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 195 T^{2} - 839 T^{3} + 195 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + T + 218 T^{2} + 126 T^{3} + 218 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 9 T + 173 T^{2} - 1606 T^{3} + 173 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 16 T + 3 p T^{2} - 2784 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 47 T^{2} - 256 T^{3} + 47 p T^{4} + p^{3} T^{6} \) |
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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
−6.82597383886115103152097747077, −6.62476670334853610059785180050, −6.50519004640076053306026871112, −6.29225074822902450509485758747, −5.94693968480935463685694953958, −5.91467031455581180554891968681, −5.75664377606120889453259489507, −5.25819526777832721885376784105, −5.11488878697100925241818021955, −5.09971396338078411086741366277, −4.61782844692307296945102744552, −4.46630307893789394838827068730, −3.91398495037311008660499534178, −3.64943228517592097414866611642, −3.51335728165785725100706819935, −3.42144629669058363788685205009, −2.92275536711509294072318885703, −2.75076857123395160049850922252, −2.68195605252461722984835362331, −2.15355087273357745853983803431, −1.65860456085650906516836146068, −1.51516477875460610649956303060, −0.63139176988533939824238816023, −0.58579249176455241752766890652, −0.38871566194525147371078479803,
0.38871566194525147371078479803, 0.58579249176455241752766890652, 0.63139176988533939824238816023, 1.51516477875460610649956303060, 1.65860456085650906516836146068, 2.15355087273357745853983803431, 2.68195605252461722984835362331, 2.75076857123395160049850922252, 2.92275536711509294072318885703, 3.42144629669058363788685205009, 3.51335728165785725100706819935, 3.64943228517592097414866611642, 3.91398495037311008660499534178, 4.46630307893789394838827068730, 4.61782844692307296945102744552, 5.09971396338078411086741366277, 5.11488878697100925241818021955, 5.25819526777832721885376784105, 5.75664377606120889453259489507, 5.91467031455581180554891968681, 5.94693968480935463685694953958, 6.29225074822902450509485758747, 6.50519004640076053306026871112, 6.62476670334853610059785180050, 6.82597383886115103152097747077
