| L(s) = 1 | + 3·2-s + 6·4-s − 2·7-s + 10·8-s − 2·11-s + 2·13-s − 6·14-s + 15·16-s − 4·17-s − 3·19-s − 6·22-s + 14·23-s + 6·26-s − 12·28-s − 14·29-s − 4·31-s + 21·32-s − 12·34-s + 17·37-s − 9·38-s + 8·41-s − 2·43-s − 12·44-s + 42·46-s + 13·47-s + 4·49-s + 12·52-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s − 0.755·7-s + 3.53·8-s − 0.603·11-s + 0.554·13-s − 1.60·14-s + 15/4·16-s − 0.970·17-s − 0.688·19-s − 1.27·22-s + 2.91·23-s + 1.17·26-s − 2.26·28-s − 2.59·29-s − 0.718·31-s + 3.71·32-s − 2.05·34-s + 2.79·37-s − 1.45·38-s + 1.24·41-s − 0.304·43-s − 1.80·44-s + 6.19·46-s + 1.89·47-s + 4/7·49-s + 1.66·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(29.30253195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(29.30253195\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T - 3 p T^{3} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 2 T + T^{2} + 20 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 2 T + 16 T^{2} - 73 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 4 T + 36 T^{2} + 143 T^{3} + 36 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 14 T + 126 T^{2} - 707 T^{3} + 126 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 128 T^{2} + 837 T^{3} + 128 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 4 T + 57 T^{2} + 96 T^{3} + 57 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 17 T + 174 T^{2} - 1249 T^{3} + 174 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 75 T^{2} - 592 T^{3} + 75 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 2 T + 89 T^{2} + 244 T^{3} + 89 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 13 T + 116 T^{2} - 697 T^{3} + 116 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 10 T + 170 T^{2} + 1057 T^{3} + 170 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 6 T + 148 T^{2} - 533 T^{3} + 148 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 22 T + 255 T^{2} - 2148 T^{3} + 255 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 70 T^{2} + 7 p T^{3} + 70 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2 T + 197 T^{2} - 260 T^{3} + 197 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 12 T + 192 T^{2} - 1509 T^{3} + 192 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 24 T + 349 T^{2} - 3472 T^{3} + 349 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 85 T^{2} - 448 T^{3} + 85 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{3} \) |
| 97 | $S_4\times C_2$ | \( 1 + 111 T^{2} - 648 T^{3} + 111 p T^{4} + p^{3} T^{6} \) |
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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
−6.79148332296788701768894668488, −6.41540425603527620442676737521, −6.41379402182036347604906268404, −6.16848512766407757022816162584, −5.87756376913086678243832051690, −5.72353319695277471114770383244, −5.43550523823473992024385017535, −5.08826779924276746590723803311, −5.04740447966574022547986681665, −4.86142866212559333207919586867, −4.47884954130528751319258693914, −4.20519562391212201786812656703, −4.06683053423261121606432552848, −3.56655545436772841553862821170, −3.54483982308777389226973747687, −3.54173208619207850515771218072, −2.91152791761070267423527016696, −2.80944422190270651768728525085, −2.46843252328081942670239562208, −2.10658192690874479963155257130, −2.00711274011848064251231028654, −1.87497544522351626053332074587, −0.855546094780470228366999836326, −0.78611990502714172025346095150, −0.66202374538604611070695406826,
0.66202374538604611070695406826, 0.78611990502714172025346095150, 0.855546094780470228366999836326, 1.87497544522351626053332074587, 2.00711274011848064251231028654, 2.10658192690874479963155257130, 2.46843252328081942670239562208, 2.80944422190270651768728525085, 2.91152791761070267423527016696, 3.54173208619207850515771218072, 3.54483982308777389226973747687, 3.56655545436772841553862821170, 4.06683053423261121606432552848, 4.20519562391212201786812656703, 4.47884954130528751319258693914, 4.86142866212559333207919586867, 5.04740447966574022547986681665, 5.08826779924276746590723803311, 5.43550523823473992024385017535, 5.72353319695277471114770383244, 5.87756376913086678243832051690, 6.16848512766407757022816162584, 6.41379402182036347604906268404, 6.41540425603527620442676737521, 6.79148332296788701768894668488
