| L(s) = 1 | + 3·3-s + 2·5-s + 3·7-s + 9-s + 7·11-s + 2·13-s + 6·15-s + 7·17-s + 3·19-s + 9·21-s − 14·23-s − 6·25-s − 7·27-s + 3·29-s + 11·31-s + 21·33-s + 6·35-s + 6·39-s − 7·41-s − 4·43-s + 2·45-s − 8·47-s + 6·49-s + 21·51-s + 53-s + 14·55-s + 9·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.894·5-s + 1.13·7-s + 1/3·9-s + 2.11·11-s + 0.554·13-s + 1.54·15-s + 1.69·17-s + 0.688·19-s + 1.96·21-s − 2.91·23-s − 6/5·25-s − 1.34·27-s + 0.557·29-s + 1.97·31-s + 3.65·33-s + 1.01·35-s + 0.960·39-s − 1.09·41-s − 0.609·43-s + 0.298·45-s − 1.16·47-s + 6/7·49-s + 2.94·51-s + 0.137·53-s + 1.88·55-s + 1.19·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(18.66560158\) |
| \(L(\frac12)\) |
\(\approx\) |
\(18.66560158\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - p T + 8 T^{2} - 14 T^{3} + 8 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 2 T + 2 p T^{2} - 18 T^{3} + 2 p^{2} T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 7 T + 4 p T^{2} - 158 T^{3} + 4 p^{2} T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 34 T^{2} - 50 T^{3} + 34 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 40 T^{2} - 132 T^{3} + 40 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 14 T + 122 T^{2} + 700 T^{3} + 122 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 14 T^{2} + 104 T^{3} + 14 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 11 T + 118 T^{2} - 698 T^{3} + 118 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 68 T^{2} - 106 T^{3} + 68 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7 T - 28 T^{2} - 424 T^{3} - 28 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 109 T^{2} + 328 T^{3} + 109 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 8 T + 112 T^{2} + 768 T^{3} + 112 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - T + 128 T^{2} - 104 T^{3} + 128 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 10 T + 178 T^{2} + 1056 T^{3} + 178 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 6 T + 134 T^{2} - 650 T^{3} + 134 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 122 T^{2} + 214 T^{3} + 122 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 152 T^{2} + 32 T^{3} + 152 p T^{4} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - T + 118 T^{2} - 244 T^{3} + 118 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 4 T + 193 T^{2} - 664 T^{3} + 193 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 31 T + 538 T^{2} - 5934 T^{3} + 538 p T^{4} - 31 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 28 T + 371 T^{2} + 3632 T^{3} + 371 p T^{4} + 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 30 T + 534 T^{2} + 6302 T^{3} + 534 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−6.98641906647958439331812663765, −6.47564461825061192249005869197, −6.40408580878679048101142994072, −6.14320317469339879104189182849, −5.94484466760484790720959921545, −5.81191300851102184630777805351, −5.70275178498974691265615399924, −5.07878989057910058875472338922, −5.00528129767450521625682881655, −4.82218904940168473079462875510, −4.39176348779579402765392290694, −4.13092749297459335052364339505, −3.92716281520920824549684092588, −3.57225496050796564723710436144, −3.45596550103930812981658628031, −3.42389240255062165013585662218, −2.82885643283701257055206870020, −2.60571594550086436799369276668, −2.52028471541153330313171010601, −1.77652584675795969726433162758, −1.76490806364060081069254543422, −1.72078824840405871527472892168, −1.39395629877196249280075932145, −0.77583049646881503805110798565, −0.52730190583695110422846800207,
0.52730190583695110422846800207, 0.77583049646881503805110798565, 1.39395629877196249280075932145, 1.72078824840405871527472892168, 1.76490806364060081069254543422, 1.77652584675795969726433162758, 2.52028471541153330313171010601, 2.60571594550086436799369276668, 2.82885643283701257055206870020, 3.42389240255062165013585662218, 3.45596550103930812981658628031, 3.57225496050796564723710436144, 3.92716281520920824549684092588, 4.13092749297459335052364339505, 4.39176348779579402765392290694, 4.82218904940168473079462875510, 5.00528129767450521625682881655, 5.07878989057910058875472338922, 5.70275178498974691265615399924, 5.81191300851102184630777805351, 5.94484466760484790720959921545, 6.14320317469339879104189182849, 6.40408580878679048101142994072, 6.47564461825061192249005869197, 6.98641906647958439331812663765
