| L(s) = 1 | + 3·3-s + 4-s + 3·5-s + 2·8-s + 6·9-s + 11-s + 3·12-s − 3·13-s + 9·15-s + 3·16-s + 17-s − 6·19-s + 3·20-s − 7·23-s + 6·24-s + 6·25-s + 10·27-s + 18·29-s − 6·31-s + 4·32-s + 3·33-s + 6·36-s + 13·37-s − 9·39-s + 6·40-s − 41-s + 44-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1/2·4-s + 1.34·5-s + 0.707·8-s + 2·9-s + 0.301·11-s + 0.866·12-s − 0.832·13-s + 2.32·15-s + 3/4·16-s + 0.242·17-s − 1.37·19-s + 0.670·20-s − 1.45·23-s + 1.22·24-s + 6/5·25-s + 1.92·27-s + 3.34·29-s − 1.07·31-s + 0.707·32-s + 0.522·33-s + 36-s + 2.13·37-s − 1.44·39-s + 0.948·40-s − 0.156·41-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 7^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(25.50943155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.50943155\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 - T^{2} - p T^{3} - p T^{4} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - T + 17 T^{2} - 38 T^{3} + 17 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 42 T^{3} + 19 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 6 T + 41 T^{2} + 164 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 7 T + 53 T^{2} + 194 T^{3} + 53 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 77 T^{2} + 340 T^{3} + 77 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 13 T + 3 p T^{2} - 646 T^{3} + 3 p^{2} T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + T + 91 T^{2} + 6 T^{3} + 91 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 17 T^{2} - 128 T^{3} + 17 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 18 T + 221 T^{2} - 1756 T^{3} + 221 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 11 T + 167 T^{2} - 1162 T^{3} + 167 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 8 T + 129 T^{2} - 816 T^{3} + 129 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 9 T + 71 T^{2} + 254 T^{3} + 71 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 4 T + 137 T^{2} - 408 T^{3} + 137 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 11 T + 237 T^{2} + 1530 T^{3} + 237 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 6 T + 119 T^{2} - 532 T^{3} + 119 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 5 T + 189 T^{2} - 854 T^{3} + 189 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 8 T + 201 T^{2} + 1200 T^{3} + 201 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 11 T + 275 T^{2} + 1954 T^{3} + 275 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 25 T + 467 T^{2} - 5094 T^{3} + 467 p T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.91493281206633399465735115589, −6.63529963581647458094148857129, −6.29197570822798414637250985516, −6.11820992547423217233706523545, −5.98513729570194234792687495831, −5.69928842820125760255693851670, −5.55650408513482868986840215105, −4.96015278803381229720416051450, −4.86400009894108297323867628836, −4.78868054004376789062905167645, −4.20215540018077232449579857630, −4.16200541165462496326483746833, −4.08486258936009998934271685154, −3.63228809030793831680528996208, −3.50423419118385080886029271900, −2.79428987210995657907625716522, −2.75713400281495733075606144378, −2.58019125330096044836527308720, −2.44163225547727784857794167658, −2.25447454491388985989428412367, −1.65534230500586758047459933751, −1.60325469636850963289917123321, −1.30622316095900928624494669560, −0.796664679697463410886830554850, −0.57328365025567604273466578201,
0.57328365025567604273466578201, 0.796664679697463410886830554850, 1.30622316095900928624494669560, 1.60325469636850963289917123321, 1.65534230500586758047459933751, 2.25447454491388985989428412367, 2.44163225547727784857794167658, 2.58019125330096044836527308720, 2.75713400281495733075606144378, 2.79428987210995657907625716522, 3.50423419118385080886029271900, 3.63228809030793831680528996208, 4.08486258936009998934271685154, 4.16200541165462496326483746833, 4.20215540018077232449579857630, 4.78868054004376789062905167645, 4.86400009894108297323867628836, 4.96015278803381229720416051450, 5.55650408513482868986840215105, 5.69928842820125760255693851670, 5.98513729570194234792687495831, 6.11820992547423217233706523545, 6.29197570822798414637250985516, 6.63529963581647458094148857129, 6.91493281206633399465735115589
