| L(s) = 1 | + 3·3-s + 6·5-s − 2·7-s + 6·9-s − 5·11-s + 18·15-s − 17-s + 7·19-s − 6·21-s + 16·25-s + 10·27-s − 2·29-s + 16·31-s − 15·33-s − 12·35-s + 22·37-s + 11·41-s + 15·43-s + 36·45-s − 7·47-s − 16·49-s − 3·51-s − 17·53-s − 30·55-s + 21·57-s + 6·59-s − 13·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2.68·5-s − 0.755·7-s + 2·9-s − 1.50·11-s + 4.64·15-s − 0.242·17-s + 1.60·19-s − 1.30·21-s + 16/5·25-s + 1.92·27-s − 0.371·29-s + 2.87·31-s − 2.61·33-s − 2.02·35-s + 3.61·37-s + 1.71·41-s + 2.28·43-s + 5.36·45-s − 1.02·47-s − 2.28·49-s − 0.420·51-s − 2.33·53-s − 4.04·55-s + 2.78·57-s + 0.781·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(22.35270987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(22.35270987\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | | \( 1 \) |
| good | 5 | $A_4\times C_2$ | \( 1 - 6 T + 4 p T^{2} - 47 T^{3} + 4 p^{2} T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $A_4\times C_2$ | \( 1 + 2 T + 20 T^{2} + 27 T^{3} + 20 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 5 T + 25 T^{2} + 69 T^{3} + 25 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 35 T^{2} + 47 T^{3} + 35 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 7 T + 71 T^{2} - 273 T^{3} + 71 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 20 T^{2} - 91 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 2 T + 72 T^{2} + 3 p T^{3} + 72 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 16 T + 134 T^{2} - 795 T^{3} + 134 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 22 T + 270 T^{2} - 2005 T^{3} + 270 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - 11 T + 147 T^{2} - 873 T^{3} + 147 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 - 15 T + 176 T^{2} - 1331 T^{3} + 176 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 7 T + 155 T^{2} + 665 T^{3} + 155 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 + 17 T + 225 T^{2} + 1761 T^{3} + 225 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 - 6 T + 161 T^{2} - 604 T^{3} + 161 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 13 T + 167 T^{2} + 1419 T^{3} + 167 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 11 T + 155 T^{2} + 1515 T^{3} + 155 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 122 T^{2} - 203 T^{3} + 122 p T^{4} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 6 T + 84 T^{2} + 47 T^{3} + 84 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 3 T + 219 T^{2} + 447 T^{3} + 219 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 12 T + 290 T^{2} + 2035 T^{3} + 290 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - T + 167 T^{2} - 65 T^{3} + 167 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + 5 T + 10 T^{2} - 667 T^{3} + 10 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.13787979668869273528889797176, −6.38443855629233597173092008936, −6.35182893571542793039384006329, −6.31378264526117820596925452133, −5.92358240890564138096217082712, −5.84994777525368762836324100569, −5.79034180529840528667960542240, −5.19918322558691087130442196889, −5.02392100430033571026840125576, −4.77660750054017168967088323643, −4.41902091258980306782709716996, −4.30370472798702039245228529742, −4.30033267057711626040557373489, −3.49011179170776805892499418437, −3.17633568766860780801573686458, −3.15926911772048801436923223337, −2.77503469413040750294299654414, −2.67972916556820626515991478310, −2.61660583421885651687757797999, −2.01918009743368334606258173453, −2.01459637560003668229707804025, −1.58879115433554025117382291857, −1.23792634125153490387343651101, −0.818535051315247205296316266835, −0.58071451581719684854969241450,
0.58071451581719684854969241450, 0.818535051315247205296316266835, 1.23792634125153490387343651101, 1.58879115433554025117382291857, 2.01459637560003668229707804025, 2.01918009743368334606258173453, 2.61660583421885651687757797999, 2.67972916556820626515991478310, 2.77503469413040750294299654414, 3.15926911772048801436923223337, 3.17633568766860780801573686458, 3.49011179170776805892499418437, 4.30033267057711626040557373489, 4.30370472798702039245228529742, 4.41902091258980306782709716996, 4.77660750054017168967088323643, 5.02392100430033571026840125576, 5.19918322558691087130442196889, 5.79034180529840528667960542240, 5.84994777525368762836324100569, 5.92358240890564138096217082712, 6.31378264526117820596925452133, 6.35182893571542793039384006329, 6.38443855629233597173092008936, 7.13787979668869273528889797176
