| L(s) = 1 | + 2-s + 3·3-s + 3·5-s + 3·6-s + 3·7-s − 8-s + 6·9-s + 3·10-s − 4·13-s + 3·14-s + 9·15-s − 3·16-s − 4·17-s + 6·18-s − 5·19-s + 9·21-s + 6·23-s − 3·24-s + 6·25-s − 4·26-s + 10·27-s + 10·29-s + 9·30-s − 31-s − 4·34-s + 9·35-s + 3·37-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1.34·5-s + 1.22·6-s + 1.13·7-s − 0.353·8-s + 2·9-s + 0.948·10-s − 1.10·13-s + 0.801·14-s + 2.32·15-s − 3/4·16-s − 0.970·17-s + 1.41·18-s − 1.14·19-s + 1.96·21-s + 1.25·23-s − 0.612·24-s + 6/5·25-s − 0.784·26-s + 1.92·27-s + 1.85·29-s + 1.64·30-s − 0.179·31-s − 0.685·34-s + 1.52·35-s + 0.493·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{3} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.22526721\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.22526721\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| good | 2 | $S_4\times C_2$ | \( 1 - T + T^{2} + p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 3 T + 11 T^{2} - 46 T^{3} + 11 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 15 T^{2} + 16 T^{3} + 15 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 44 T^{2} + 122 T^{3} + 44 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 5 T + 53 T^{2} + 174 T^{3} + 53 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 6 T + 68 T^{2} - 242 T^{3} + 68 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 10 T + 99 T^{2} - 564 T^{3} + 99 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + T + 64 T^{2} + T^{3} + 64 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 3 T - 3 T^{2} + 326 T^{3} - 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 10 T + 71 T^{2} + 428 T^{3} + 71 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 2 T + 109 T^{2} - 140 T^{3} + 109 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 172 T^{2} - 1212 T^{3} + 172 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 20 T^{2} - 586 T^{3} + 20 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 18 T + 149 T^{2} - 1036 T^{3} + 149 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 27 T + 374 T^{2} - 3427 T^{3} + 374 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + T + 147 T^{2} - 38 T^{3} + 147 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 14 T + 61 T^{2} + 156 T^{3} + 61 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 7 T + 223 T^{2} + 1006 T^{3} + 223 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 9 T + 152 T^{2} - 1505 T^{3} + 152 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 89 | $S_4\times C_2$ | \( 1 - 32 T + 587 T^{2} - 6664 T^{3} + 587 p T^{4} - 32 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - T + 237 T^{2} - 22 T^{3} + 237 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.329732931410157620358794860031, −7.958971128028535731626569660208, −7.901849471184717676165620497589, −7.44234122427314567910962518395, −7.00883826678927327401722549710, −6.87504369543183329790390701418, −6.63704595542074971390252223261, −6.53797220438516698261484925717, −6.15169859742759891738920122940, −5.62531755212223775048361122898, −5.24225845762575283357219830946, −5.10314872940223102064349527749, −5.01730094311932996264547549612, −4.50778713356136487483364021916, −4.26611802124737412364292139433, −4.19250654409666193645143578329, −3.53598967052914342013638198294, −3.43373491291851931288478261464, −2.78516966713804589739700595851, −2.58368507917262678477986150671, −2.24204098553901254426500093061, −2.13136431246681253331907338546, −1.91328413333990114939184072882, −1.00717342677188932063711167315, −0.824133369501425312176330807248,
0.824133369501425312176330807248, 1.00717342677188932063711167315, 1.91328413333990114939184072882, 2.13136431246681253331907338546, 2.24204098553901254426500093061, 2.58368507917262678477986150671, 2.78516966713804589739700595851, 3.43373491291851931288478261464, 3.53598967052914342013638198294, 4.19250654409666193645143578329, 4.26611802124737412364292139433, 4.50778713356136487483364021916, 5.01730094311932996264547549612, 5.10314872940223102064349527749, 5.24225845762575283357219830946, 5.62531755212223775048361122898, 6.15169859742759891738920122940, 6.53797220438516698261484925717, 6.63704595542074971390252223261, 6.87504369543183329790390701418, 7.00883826678927327401722549710, 7.44234122427314567910962518395, 7.901849471184717676165620497589, 7.958971128028535731626569660208, 8.329732931410157620358794860031
