| L(s) = 1 | + 3·3-s + 3·5-s − 3·7-s + 8-s + 6·9-s + 6·11-s + 9·13-s + 9·15-s + 6·17-s − 6·19-s − 9·21-s − 3·23-s + 3·24-s − 3·25-s + 10·27-s + 6·29-s − 18·31-s + 18·33-s − 9·35-s + 6·37-s + 27·39-s + 3·40-s − 6·41-s − 9·43-s + 18·45-s + 18·47-s + 6·49-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.34·5-s − 1.13·7-s + 0.353·8-s + 2·9-s + 1.80·11-s + 2.49·13-s + 2.32·15-s + 1.45·17-s − 1.37·19-s − 1.96·21-s − 0.625·23-s + 0.612·24-s − 3/5·25-s + 1.92·27-s + 1.11·29-s − 3.23·31-s + 3.13·33-s − 1.52·35-s + 0.986·37-s + 4.32·39-s + 0.474·40-s − 0.937·41-s − 1.37·43-s + 2.68·45-s + 2.62·47-s + 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.486027069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.486027069\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{3} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 3 T + 12 T^{2} - 24 T^{3} + 12 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 30 T^{2} - 112 T^{3} + 30 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 9 T + 54 T^{2} - 240 T^{3} + 54 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 154 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 54 T^{2} + 208 T^{3} + 54 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 6 T + 42 T^{2} - 86 T^{3} + 42 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 18 T + 6 p T^{2} + 1244 T^{3} + 6 p^{2} T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 6 T + 66 T^{2} - 494 T^{3} + 66 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 30 T^{2} - 98 T^{3} + 30 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 9 T + 126 T^{2} + 650 T^{3} + 126 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 201 T^{2} - 1500 T^{3} + 201 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 3 T + 156 T^{2} - 312 T^{3} + 156 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 3 T + 120 T^{2} - 342 T^{3} + 120 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 3 T + 156 T^{2} + 276 T^{3} + 156 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 21 T + 228 T^{2} + 1906 T^{3} + 228 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 9 T + 90 T^{2} - 854 T^{3} + 90 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 192 T^{2} - 1762 T^{3} + 192 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 12 T + 228 T^{2} - 1576 T^{3} + 228 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 276 T^{2} - 1956 T^{3} + 276 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 15 T + 330 T^{2} - 2720 T^{3} + 330 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 264 T^{2} + 1846 T^{3} + 264 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−9.795116117014375684275452146333, −9.197089042058381749106935282653, −9.124858598744623739981413344820, −9.116418221921225878490914229367, −8.822960260355498716314835112257, −8.378429271904773033536201663148, −8.042279318630985972064554862880, −7.66848761868815121323227590754, −7.50686441972288628982024985433, −6.94633823164937920862633001278, −6.56624117934102300616147899820, −6.31222118796348718741519094840, −6.16203618332346767751848145449, −5.88928690736586041659126863633, −5.38964569241539682853565737802, −5.02031126817553822889747265920, −4.02610935804310561791245665266, −4.00068566652171835500489046491, −3.86323176938113874084946195290, −3.41719692234138307741426808564, −3.13484636625349900617561674732, −2.23578913682427248025091532263, −2.14123424448105500716198057673, −1.39431316014339394278105057010, −1.27326967760055291313327154889,
1.27326967760055291313327154889, 1.39431316014339394278105057010, 2.14123424448105500716198057673, 2.23578913682427248025091532263, 3.13484636625349900617561674732, 3.41719692234138307741426808564, 3.86323176938113874084946195290, 4.00068566652171835500489046491, 4.02610935804310561791245665266, 5.02031126817553822889747265920, 5.38964569241539682853565737802, 5.88928690736586041659126863633, 6.16203618332346767751848145449, 6.31222118796348718741519094840, 6.56624117934102300616147899820, 6.94633823164937920862633001278, 7.50686441972288628982024985433, 7.66848761868815121323227590754, 8.042279318630985972064554862880, 8.378429271904773033536201663148, 8.822960260355498716314835112257, 9.116418221921225878490914229367, 9.124858598744623739981413344820, 9.197089042058381749106935282653, 9.795116117014375684275452146333
