| L(s) = 1 | + 3·2-s + 6·4-s + 2·5-s − 3·7-s + 10·8-s + 6·10-s + 6·11-s + 8·13-s − 9·14-s + 15·16-s − 3·19-s + 12·20-s + 18·22-s + 4·23-s + 25-s + 24·26-s − 18·28-s + 2·29-s − 2·31-s + 21·32-s − 6·35-s + 4·37-s − 9·38-s + 20·40-s + 2·41-s − 4·43-s + 36·44-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3·4-s + 0.894·5-s − 1.13·7-s + 3.53·8-s + 1.89·10-s + 1.80·11-s + 2.21·13-s − 2.40·14-s + 15/4·16-s − 0.688·19-s + 2.68·20-s + 3.83·22-s + 0.834·23-s + 1/5·25-s + 4.70·26-s − 3.40·28-s + 0.371·29-s − 0.359·31-s + 3.71·32-s − 1.01·35-s + 0.657·37-s − 1.45·38-s + 3.16·40-s + 0.312·41-s − 0.609·43-s + 5.42·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 7^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(28.58812526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(28.58812526\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 5 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 6 T + 29 T^{2} - 92 T^{3} + 29 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 8 T + 47 T^{2} - 192 T^{3} + 47 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 35 T^{2} - 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 4 T + 61 T^{2} - 168 T^{3} + 61 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $D_{6}$ | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 2 T + 73 T^{2} + 116 T^{3} + 73 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 4 T + 95 T^{2} - 264 T^{3} + 95 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 2 T + 71 T^{2} - 124 T^{3} + 71 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 4 T + 49 T^{2} + 280 T^{3} + 49 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 16 T + 189 T^{2} - 1536 T^{3} + 189 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 61 | $S_4\times C_2$ | \( 1 - 10 T + 131 T^{2} - 1020 T^{3} + 131 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 185 T^{2} + 16 T^{3} + 185 p T^{4} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 24 T + 341 T^{2} - 3280 T^{3} + 341 p T^{4} - 24 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 6 T + 167 T^{2} + 628 T^{3} + 167 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 18 T + 113 T^{2} + 308 T^{3} + 113 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 14 T + 261 T^{2} - 2172 T^{3} + 261 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 18 T + 311 T^{2} - 3164 T^{3} + 311 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 2 T + 207 T^{2} + 620 T^{3} + 207 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.049589851917529919871114972992, −7.42951301828567622090344125061, −7.16282715987046199897109426834, −6.89055091181076924624750752190, −6.77237520759666273859463781381, −6.53914857205557580946783640968, −6.32200208763900613786059847005, −5.93617529873925963653732477395, −5.89138505836690521007590508296, −5.75270634618163321800799109803, −5.35120023269089736971651084965, −5.01999485051464308101566694265, −4.71676662908227033220117105892, −4.15625128870022072196939321939, −4.07564016608208272317300307096, −4.01236380573935747466359478896, −3.52074253607213286519455799123, −3.31777824065216942327734139276, −3.18622998278885204491576428880, −2.43703269449703157678076425390, −2.40563849250399938361963359014, −2.02999430096934033186869330473, −1.45563058974872740056334224117, −1.04598084507985906819486555199, −0.868875398972721218198239553538,
0.868875398972721218198239553538, 1.04598084507985906819486555199, 1.45563058974872740056334224117, 2.02999430096934033186869330473, 2.40563849250399938361963359014, 2.43703269449703157678076425390, 3.18622998278885204491576428880, 3.31777824065216942327734139276, 3.52074253607213286519455799123, 4.01236380573935747466359478896, 4.07564016608208272317300307096, 4.15625128870022072196939321939, 4.71676662908227033220117105892, 5.01999485051464308101566694265, 5.35120023269089736971651084965, 5.75270634618163321800799109803, 5.89138505836690521007590508296, 5.93617529873925963653732477395, 6.32200208763900613786059847005, 6.53914857205557580946783640968, 6.77237520759666273859463781381, 6.89055091181076924624750752190, 7.16282715987046199897109426834, 7.42951301828567622090344125061, 8.049589851917529919871114972992
