| L(s) = 1 | + 3·3-s − 8-s + 6·9-s − 3·11-s + 12·13-s − 6·17-s − 3·19-s − 9·23-s − 3·24-s + 10·27-s − 15·29-s + 15·31-s − 9·33-s + 12·37-s + 36·39-s + 12·41-s + 12·43-s + 12·47-s − 18·51-s − 9·53-s − 9·57-s + 12·59-s + 3·61-s − 7·64-s + 15·67-s − 27·69-s + 18·71-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.353·8-s + 2·9-s − 0.904·11-s + 3.32·13-s − 1.45·17-s − 0.688·19-s − 1.87·23-s − 0.612·24-s + 1.92·27-s − 2.78·29-s + 2.69·31-s − 1.56·33-s + 1.97·37-s + 5.76·39-s + 1.87·41-s + 1.82·43-s + 1.75·47-s − 2.52·51-s − 1.23·53-s − 1.19·57-s + 1.56·59-s + 0.384·61-s − 7/8·64-s + 1.83·67-s − 3.25·69-s + 2.13·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.798818779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.798818779\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 2 | $D_{6}$ | \( 1 + T^{3} + p^{3} T^{6} \) |
| 7 | $D_{6}$ | \( 1 + 16 T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 21 T^{2} + 50 T^{3} + 21 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 17 | $S_4\times C_2$ | \( 1 + 6 T + 15 T^{2} - 4 T^{3} + 15 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 9 T + 75 T^{2} + 362 T^{3} + 75 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{3} \) |
| 31 | $S_4\times C_2$ | \( 1 - 15 T + 153 T^{2} - 978 T^{3} + 153 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 12 T + 135 T^{2} - 864 T^{3} + 135 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 12 T + 156 T^{2} - 990 T^{3} + 156 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 47 | $S_4\times C_2$ | \( 1 - 12 T + 129 T^{2} - 936 T^{3} + 129 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 90 T^{2} + 757 T^{3} + 90 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 12 T + 168 T^{2} - 1096 T^{3} + 168 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 3 T + 126 T^{2} - 323 T^{3} + 126 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 15 T + 261 T^{2} - 2062 T^{3} + 261 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 18 T + 264 T^{2} - 2274 T^{3} + 264 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 126 T^{2} - 407 T^{3} + 126 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 3 T + 51 T^{2} + 146 T^{3} + 51 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 3 T + 237 T^{2} - 486 T^{3} + 237 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 3 T + 186 T^{2} + 579 T^{3} + 186 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 12 T + 219 T^{2} + 1424 T^{3} + 219 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.465419334106248312860977439940, −8.228814813891283501026526871689, −8.019348638575474601586213441489, −7.78095851205330655340080988979, −7.67665811955411065193849233090, −7.26176804995865249815114508735, −6.70833599322353207277584593512, −6.56035768708430895163065137986, −6.22732384503522644621580693384, −6.03905004688767047043821028064, −5.92691094009755233414175935842, −5.30903745077382647517845232031, −5.25175240876527058877968695226, −4.42253330885857925496930507112, −4.10477243057458359655380361046, −4.07028220141468214138204329344, −3.93396579069535076628270207304, −3.64953897575663773561367134396, −3.00374812762937519308403629602, −2.65907996937476021672666361868, −2.37685323714405405534974646639, −2.22977262124357249504897276509, −1.68212040412198250094898676165, −1.07532643892757758596448616728, −0.66665524101752809666315282722,
0.66665524101752809666315282722, 1.07532643892757758596448616728, 1.68212040412198250094898676165, 2.22977262124357249504897276509, 2.37685323714405405534974646639, 2.65907996937476021672666361868, 3.00374812762937519308403629602, 3.64953897575663773561367134396, 3.93396579069535076628270207304, 4.07028220141468214138204329344, 4.10477243057458359655380361046, 4.42253330885857925496930507112, 5.25175240876527058877968695226, 5.30903745077382647517845232031, 5.92691094009755233414175935842, 6.03905004688767047043821028064, 6.22732384503522644621580693384, 6.56035768708430895163065137986, 6.70833599322353207277584593512, 7.26176804995865249815114508735, 7.67665811955411065193849233090, 7.78095851205330655340080988979, 8.019348638575474601586213441489, 8.228814813891283501026526871689, 8.465419334106248312860977439940
