| L(s) = 1 | + 3·5-s + 5·7-s − 3·11-s + 3·13-s − 17-s + 4·19-s − 5·23-s + 6·25-s + 4·29-s + 10·31-s + 15·35-s + 5·37-s − 5·41-s + 12·43-s − 2·47-s + 10·49-s + 13·53-s − 9·55-s + 21·61-s + 9·65-s + 8·67-s + 71-s + 28·73-s − 15·77-s − 21·79-s − 4·83-s − 3·85-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.88·7-s − 0.904·11-s + 0.832·13-s − 0.242·17-s + 0.917·19-s − 1.04·23-s + 6/5·25-s + 0.742·29-s + 1.79·31-s + 2.53·35-s + 0.821·37-s − 0.780·41-s + 1.82·43-s − 0.291·47-s + 10/7·49-s + 1.78·53-s − 1.21·55-s + 2.68·61-s + 1.11·65-s + 0.977·67-s + 0.118·71-s + 3.27·73-s − 1.70·77-s − 2.36·79-s − 0.439·83-s − 0.325·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.44593256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.44593256\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 7 | $A_4\times C_2$ | \( 1 - 5 T + 15 T^{2} - 38 T^{3} + 15 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_6$ | \( 1 + 3 T - 7 T^{2} - 62 T^{3} - 7 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + T + 37 T^{2} + 42 T^{3} + 37 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 - 4 T + 5 T^{2} + 24 T^{3} + 5 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $A_4\times C_2$ | \( 1 + 5 T + 63 T^{2} + 198 T^{3} + 63 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 - 4 T + 35 T^{2} - 56 T^{3} + 35 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 - 10 T + 69 T^{2} - 364 T^{3} + 69 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 - 5 T + 19 T^{2} + 178 T^{3} + 19 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 + 5 T + 31 T^{2} - 138 T^{3} + 31 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 47 | $A_4\times C_2$ | \( 1 + 2 T + 85 T^{2} + 252 T^{3} + 85 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 13 T + 115 T^{2} - 894 T^{3} + 115 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 61 | $A_4\times C_2$ | \( 1 - 21 T + 287 T^{2} - 2518 T^{3} + 287 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 - 8 T - 7 T^{2} + 336 T^{3} - 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 - T + 113 T^{2} - 494 T^{3} + 113 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 28 T + 423 T^{2} - 4264 T^{3} + 423 p T^{4} - 28 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 21 T + 341 T^{2} + 3446 T^{3} + 341 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 + 4 T + 25 T^{2} + 1176 T^{3} + 25 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 + 11 T + 35 T^{2} - 638 T^{3} + 35 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 - 25 T + 5 p T^{2} - 5322 T^{3} + 5 p^{2} T^{4} - 25 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.50328401201518424574131474747, −6.95081247507355643078917388942, −6.93357279396993068278771850397, −6.62895053304988908201674097988, −6.34418211677649098240280757299, −5.97834159388555341701342232441, −5.91484863612831893657472966847, −5.52778262827567581930659220682, −5.33336971276553962236628581690, −5.29326176529849271978199074445, −4.77027287288402940525542307607, −4.71424156803707491662224941703, −4.48524509190232527954781336997, −3.95654699648626685938149793781, −3.89404716041836738850998593859, −3.59536879944592642442972407312, −2.89478982675418459793522225020, −2.89076283881792973530101427080, −2.48417913635368069224187094256, −2.14078039091136155888884567708, −2.06490471596421162345683651564, −1.64050586077519938584336285424, −1.14062991009885315502905514346, −0.874987811117174738610411270047, −0.66421042371712788949939497322,
0.66421042371712788949939497322, 0.874987811117174738610411270047, 1.14062991009885315502905514346, 1.64050586077519938584336285424, 2.06490471596421162345683651564, 2.14078039091136155888884567708, 2.48417913635368069224187094256, 2.89076283881792973530101427080, 2.89478982675418459793522225020, 3.59536879944592642442972407312, 3.89404716041836738850998593859, 3.95654699648626685938149793781, 4.48524509190232527954781336997, 4.71424156803707491662224941703, 4.77027287288402940525542307607, 5.29326176529849271978199074445, 5.33336971276553962236628581690, 5.52778262827567581930659220682, 5.91484863612831893657472966847, 5.97834159388555341701342232441, 6.34418211677649098240280757299, 6.62895053304988908201674097988, 6.93357279396993068278771850397, 6.95081247507355643078917388942, 7.50328401201518424574131474747
