| L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 3·7-s − 10·8-s + 9-s + 3·11-s − 6·12-s + 13-s + 9·14-s + 15·16-s + 7·17-s − 3·18-s + 3·19-s + 3·21-s − 9·22-s + 3·23-s + 10·24-s − 3·26-s + 5·27-s − 18·28-s − 4·29-s − 5·31-s − 21·32-s − 3·33-s − 21·34-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 3·4-s + 1.22·6-s − 1.13·7-s − 3.53·8-s + 1/3·9-s + 0.904·11-s − 1.73·12-s + 0.277·13-s + 2.40·14-s + 15/4·16-s + 1.69·17-s − 0.707·18-s + 0.688·19-s + 0.654·21-s − 1.91·22-s + 0.625·23-s + 2.04·24-s − 0.588·26-s + 0.962·27-s − 3.40·28-s − 0.742·29-s − 0.898·31-s − 3.71·32-s − 0.522·33-s − 3.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7113461557\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7113461557\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + T - 2 p T^{3} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T - 22 T^{3} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 3 T - 6 T^{2} + 78 T^{3} - 6 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - T + 24 T^{2} - 8 T^{3} + 24 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 58 T^{2} - 220 T^{3} + 58 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 3 T + 36 T^{2} - 50 T^{3} + 36 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 4 T + 55 T^{2} + 256 T^{3} + 55 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 5 T + 86 T^{2} + 302 T^{3} + 86 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 116 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - T + 64 T^{2} + 104 T^{3} + 64 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 14 T + 145 T^{2} - 1028 T^{3} + 145 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 59 | $S_4\times C_2$ | \( 1 - 14 T + 205 T^{2} - 1508 T^{3} + 205 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - T + 26 T^{2} + 404 T^{3} + 26 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $D_{6}$ | \( 1 + 8 T + 57 T^{2} + 688 T^{3} + 57 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 11 T + 244 T^{2} - 1586 T^{3} + 244 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 8 T + 179 T^{2} - 920 T^{3} + 179 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 4 T - 3 T^{2} - 520 T^{3} - 3 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 229 T^{2} + 1232 T^{3} + 229 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 219 T^{2} - 2052 T^{3} + 219 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 33 T + 570 T^{2} - 6568 T^{3} + 570 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.832790662230635775275021571195, −8.477104520479873143897016174306, −8.297421217678608371078629272566, −7.984596621250526134788184652866, −7.44744607791650724545364265928, −7.37338431189055092303694247480, −7.17446505293404682935345569038, −6.84085879922185757567362027017, −6.52023634843169657001551212315, −6.47802665037519701014282471412, −5.93480342872570257463588560002, −5.60229015874661185041588346918, −5.56994526733282173525839840899, −5.03869117141804292781032290291, −4.85699245367611369076996366602, −3.89377024738737504773973729519, −3.85209003540653046182139281946, −3.50995800773118232359249925422, −3.24168984272545538106909163075, −2.55109832646693204624787408112, −2.50908549184883622448123360561, −1.76661072126911022167516583972, −1.25111557919532713809209952606, −0.988449079867128240513664410925, −0.47417773513251705065752782599,
0.47417773513251705065752782599, 0.988449079867128240513664410925, 1.25111557919532713809209952606, 1.76661072126911022167516583972, 2.50908549184883622448123360561, 2.55109832646693204624787408112, 3.24168984272545538106909163075, 3.50995800773118232359249925422, 3.85209003540653046182139281946, 3.89377024738737504773973729519, 4.85699245367611369076996366602, 5.03869117141804292781032290291, 5.56994526733282173525839840899, 5.60229015874661185041588346918, 5.93480342872570257463588560002, 6.47802665037519701014282471412, 6.52023634843169657001551212315, 6.84085879922185757567362027017, 7.17446505293404682935345569038, 7.37338431189055092303694247480, 7.44744607791650724545364265928, 7.984596621250526134788184652866, 8.297421217678608371078629272566, 8.477104520479873143897016174306, 8.832790662230635775275021571195
