| L(s) = 1 | + 3·5-s − 3·11-s + 3·13-s + 6·17-s − 3·19-s − 3·23-s + 6·25-s + 6·27-s + 12·29-s − 12·31-s + 9·37-s + 9·41-s − 12·43-s + 15·47-s + 9·53-s − 9·55-s − 24·59-s − 6·61-s + 9·65-s − 6·67-s + 18·79-s + 30·83-s + 18·85-s − 9·95-s − 6·97-s − 18·101-s + 6·103-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.904·11-s + 0.832·13-s + 1.45·17-s − 0.688·19-s − 0.625·23-s + 6/5·25-s + 1.15·27-s + 2.22·29-s − 2.15·31-s + 1.47·37-s + 1.40·41-s − 1.82·43-s + 2.18·47-s + 1.23·53-s − 1.21·55-s − 3.12·59-s − 0.768·61-s + 1.11·65-s − 0.733·67-s + 2.02·79-s + 3.29·83-s + 1.95·85-s − 0.923·95-s − 0.609·97-s − 1.79·101-s + 0.591·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.129277251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.129277251\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{6}$ | \( 1 - 2 p T^{3} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 9 T^{2} + 2 p T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 3 T + 15 T^{2} - 10 T^{3} + 15 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 33 T^{2} + 130 T^{3} + 33 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 3 T + 54 T^{2} + 145 T^{3} + 54 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 12 T + 108 T^{2} - 670 T^{3} + 108 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 12 T + 105 T^{2} + 616 T^{3} + 105 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 9 T + 3 p T^{2} - 570 T^{3} + 3 p^{2} T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 357 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 12 T + 168 T^{2} + 1054 T^{3} + 168 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 15 T + 45 T^{2} + 178 T^{3} + 45 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 9 T + 87 T^{2} - 330 T^{3} + 87 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 + 6 T + 96 T^{2} + 188 T^{3} + 96 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 6 T + 132 T^{2} + 812 T^{3} + 132 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 73 | $S_4\times C_2$ | \( 1 + 111 T^{2} + 336 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 18 T + 3 p T^{2} - 2076 T^{3} + 3 p^{2} T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 30 T + 540 T^{2} - 5884 T^{3} + 540 p T^{4} - 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 240 T^{2} + 42 T^{3} + 240 p T^{4} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
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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
−7.42963365046289181360739133812, −7.41196237451202458043486298199, −6.98065258325323598835706676713, −6.62625753395799793470391098210, −6.35164053398424467537779099339, −6.30768045483592536595812878848, −6.05482025874216854505586192622, −5.65346600492019728779532380201, −5.56358841414786727294997622767, −5.41360163515490684065624225217, −4.87893606202459111163859542179, −4.78841514734820239598964125333, −4.50744971005793161169550870753, −4.20541931634667915487115131402, −3.75311516421018317589868654842, −3.61834297070763244105228041787, −3.12207414201594115168810450213, −2.81977670380054504331876735560, −2.79780257158742548072306124190, −2.33357268417425751637310183290, −1.86965615647717902189374671041, −1.79027217306870455025806794431, −1.19227825887847293890009198546, −0.885136737516375528682617164397, −0.50263375601897691896582232332,
0.50263375601897691896582232332, 0.885136737516375528682617164397, 1.19227825887847293890009198546, 1.79027217306870455025806794431, 1.86965615647717902189374671041, 2.33357268417425751637310183290, 2.79780257158742548072306124190, 2.81977670380054504331876735560, 3.12207414201594115168810450213, 3.61834297070763244105228041787, 3.75311516421018317589868654842, 4.20541931634667915487115131402, 4.50744971005793161169550870753, 4.78841514734820239598964125333, 4.87893606202459111163859542179, 5.41360163515490684065624225217, 5.56358841414786727294997622767, 5.65346600492019728779532380201, 6.05482025874216854505586192622, 6.30768045483592536595812878848, 6.35164053398424467537779099339, 6.62625753395799793470391098210, 6.98065258325323598835706676713, 7.41196237451202458043486298199, 7.42963365046289181360739133812
