| L(s) = 1 | + 3-s − 2·5-s − 4·9-s + 3·11-s − 11·13-s − 2·15-s − 3·17-s + 11·19-s − 12·23-s − 4·25-s − 6·27-s − 9·29-s + 3·31-s + 3·33-s − 4·37-s − 11·39-s − 5·41-s − 2·43-s + 8·45-s + 3·47-s − 3·51-s + 17·53-s − 6·55-s + 11·57-s − 8·59-s − 24·61-s + 22·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 4/3·9-s + 0.904·11-s − 3.05·13-s − 0.516·15-s − 0.727·17-s + 2.52·19-s − 2.50·23-s − 4/5·25-s − 1.15·27-s − 1.67·29-s + 0.538·31-s + 0.522·33-s − 0.657·37-s − 1.76·39-s − 0.780·41-s − 0.304·43-s + 1.19·45-s + 0.437·47-s − 0.420·51-s + 2.33·53-s − 0.809·55-s + 1.45·57-s − 1.04·59-s − 3.07·61-s + 2.72·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 7^{6} \cdot 11^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - T + 5 T^{2} - p T^{3} + 5 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 2 T + 8 T^{2} + 3 p T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 11 T + 75 T^{2} + 321 T^{3} + 75 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 49 T^{2} + 95 T^{3} + 49 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 11 T + 77 T^{2} - p^{2} T^{3} + 77 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 12 T + 112 T^{2} + 599 T^{3} + 112 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 9 T + 67 T^{2} + 469 T^{3} + 67 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 3 T + 49 T^{2} - 79 T^{3} + 49 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 4 T + 75 T^{2} + 144 T^{3} + 75 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 5 T + 43 T^{2} + 301 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 104 T^{2} + 131 T^{3} + 104 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 3 T + 139 T^{2} - 275 T^{3} + 139 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 17 T + 233 T^{2} - 1823 T^{3} + 233 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 20 T^{2} - 379 T^{3} + 20 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 355 T^{2} + 3304 T^{3} + 355 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 269 T^{2} - 2216 T^{3} + 269 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 7 T + 127 T^{2} + 575 T^{3} + 127 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 20 T + 244 T^{2} + 2295 T^{3} + 244 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 3 T + 199 T^{2} + 333 T^{3} + 199 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 11 T + 265 T^{2} - 1823 T^{3} + 265 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - T + 259 T^{2} - 175 T^{3} + 259 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 279 T^{2} - 1699 T^{3} + 279 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.32255429811050133652399069427, −7.10364298926763192359338759541, −6.78094595729083193335853895353, −6.74881824547214471158115257167, −6.16197851724481816273386821610, −5.94065513104705006758348180885, −5.90781834920432942799559632106, −5.46368976807143408589370787211, −5.46263130391021075909639756851, −5.14617601375481145809073928517, −4.79617934412699436339602122725, −4.69156912134922206940970029770, −4.23316110447534602139615908279, −4.09223180995199490008784113027, −3.90241172208552138177481575831, −3.64350303382618748926062735703, −3.17549620372375255231116719338, −3.04769882417094703490581601776, −3.01748196908401718506560306975, −2.33302853413299642984041036805, −2.25128378077648748407959038737, −2.19174538134469588899435708181, −1.51471020276721031040875107841, −1.49162625882582470249531094170, −0.74792191692199101347836075508, 0, 0, 0,
0.74792191692199101347836075508, 1.49162625882582470249531094170, 1.51471020276721031040875107841, 2.19174538134469588899435708181, 2.25128378077648748407959038737, 2.33302853413299642984041036805, 3.01748196908401718506560306975, 3.04769882417094703490581601776, 3.17549620372375255231116719338, 3.64350303382618748926062735703, 3.90241172208552138177481575831, 4.09223180995199490008784113027, 4.23316110447534602139615908279, 4.69156912134922206940970029770, 4.79617934412699436339602122725, 5.14617601375481145809073928517, 5.46263130391021075909639756851, 5.46368976807143408589370787211, 5.90781834920432942799559632106, 5.94065513104705006758348180885, 6.16197851724481816273386821610, 6.74881824547214471158115257167, 6.78094595729083193335853895353, 7.10364298926763192359338759541, 7.32255429811050133652399069427
