| L(s) = 1 | + 3·2-s + 6·4-s − 3·5-s + 6·7-s + 10·8-s − 3·9-s − 9·10-s + 18·14-s + 15·16-s + 9·17-s − 9·18-s + 3·19-s − 18·20-s − 15·23-s + 3·25-s + 3·27-s + 36·28-s − 6·29-s − 3·31-s + 21·32-s + 27·34-s − 18·35-s − 18·36-s − 6·37-s + 9·38-s − 30·40-s + 9·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 1.34·5-s + 2.26·7-s + 3.53·8-s − 9-s − 2.84·10-s + 4.81·14-s + 15/4·16-s + 2.18·17-s − 2.12·18-s + 0.688·19-s − 4.02·20-s − 3.12·23-s + 3/5·25-s + 0.577·27-s + 6.80·28-s − 1.11·29-s − 0.538·31-s + 3.71·32-s + 4.63·34-s − 3.04·35-s − 3·36-s − 0.986·37-s + 1.45·38-s − 4.74·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 11^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.44085848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.44085848\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} - p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 + 3 T + 6 T^{2} + 12 T^{3} + 6 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 6 T + 15 T^{2} - 27 T^{3} + 15 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 9 T + 69 T^{2} - 302 T^{3} + 69 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 15 T + 135 T^{2} + 766 T^{3} + 135 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 201 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 3 T + 42 T^{2} + 52 T^{3} + 42 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 87 T^{2} + 348 T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 9 T + 78 T^{2} - 616 T^{3} + 78 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 21 T + 222 T^{2} - 1622 T^{3} + 222 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 12 T + 81 T^{2} + 456 T^{3} + 81 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 9 T + 69 T^{2} + 502 T^{3} + 69 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 3 T + 153 T^{2} + 290 T^{3} + 153 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 24 T + 339 T^{2} + 3120 T^{3} + 339 p T^{4} + 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 105 T^{2} + 123 T^{3} + 105 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 21 T + 216 T^{2} - 1676 T^{3} + 216 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 3 T + 213 T^{2} - 426 T^{3} + 213 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 4 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 33 T + 594 T^{2} - 6582 T^{3} + 594 p T^{4} - 33 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 6 T + 231 T^{2} - 924 T^{3} + 231 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 12 T + 195 T^{2} - 2072 T^{3} + 195 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.47089459622548593629933736718, −7.44922966589162054389085277794, −6.74213818813617001930405869079, −6.48597791068221036271984632331, −6.11913326865801565796274722135, −6.03483175898876479167646384296, −5.83525763078345636404303867827, −5.59083887467222899919622946134, −5.27136964642923887326879059357, −4.99235116511441522350164379501, −4.91205055618986485109982935396, −4.69687699668499012400286236153, −4.29817108437343017689720436166, −3.92345421645479421254778259783, −3.80101491747789085793622769874, −3.79734070238686811712256055200, −3.22074542336254436137027349507, −3.17069275329707419667172218240, −2.76952688528280443439778794305, −2.18174764107316283066687412850, −2.00291871005991870335858583302, −1.97617452421397976732441570666, −1.18696682990960245088344180535, −1.14323486462799679468290454975, −0.38907638635255401009910198760,
0.38907638635255401009910198760, 1.14323486462799679468290454975, 1.18696682990960245088344180535, 1.97617452421397976732441570666, 2.00291871005991870335858583302, 2.18174764107316283066687412850, 2.76952688528280443439778794305, 3.17069275329707419667172218240, 3.22074542336254436137027349507, 3.79734070238686811712256055200, 3.80101491747789085793622769874, 3.92345421645479421254778259783, 4.29817108437343017689720436166, 4.69687699668499012400286236153, 4.91205055618986485109982935396, 4.99235116511441522350164379501, 5.27136964642923887326879059357, 5.59083887467222899919622946134, 5.83525763078345636404303867827, 6.03483175898876479167646384296, 6.11913326865801565796274722135, 6.48597791068221036271984632331, 6.74213818813617001930405869079, 7.44922966589162054389085277794, 7.47089459622548593629933736718
