| L(s) = 1 | − 3·5-s − 3·7-s − 3·9-s − 3·11-s − 3·17-s − 3·19-s − 3·23-s + 6·25-s − 3·27-s + 3·29-s − 6·31-s + 9·35-s + 12·37-s − 6·41-s − 18·43-s + 9·45-s − 15·47-s − 9·49-s + 6·53-s + 9·55-s − 6·59-s + 27·61-s + 9·63-s − 12·67-s − 6·71-s − 15·73-s + 9·77-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.13·7-s − 9-s − 0.904·11-s − 0.727·17-s − 0.688·19-s − 0.625·23-s + 6/5·25-s − 0.577·27-s + 0.557·29-s − 1.07·31-s + 1.52·35-s + 1.97·37-s − 0.937·41-s − 2.74·43-s + 1.34·45-s − 2.18·47-s − 9/7·49-s + 0.824·53-s + 1.21·55-s − 0.781·59-s + 3.45·61-s + 1.13·63-s − 1.46·67-s − 0.712·71-s − 1.75·73-s + 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 23 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 40 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 3 T + 18 T^{2} + 78 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 21 T^{2} + 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 3 T + 36 T^{2} + 114 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 3 T + 24 T^{2} + 162 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 3 T + 81 T^{2} - 162 T^{3} + 81 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 6 T + 51 T^{2} + 123 T^{3} + 51 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{3} \) |
| 41 | $S_4\times C_2$ | \( 1 + 6 T + 63 T^{2} + 423 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 18 T + 201 T^{2} + 1516 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 15 T + 207 T^{2} + 1486 T^{3} + 207 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 27 T^{2} - 100 T^{3} + 27 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 6 T + 141 T^{2} + 676 T^{3} + 141 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 27 T + 408 T^{2} - 3890 T^{3} + 408 p T^{4} - 27 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 12 T - 39 T^{2} - 1336 T^{3} - 39 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 6 T + 153 T^{2} + 649 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 15 T + 177 T^{2} + 1602 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 83 | $S_4\times C_2$ | \( 1 - 12 T + 189 T^{2} - 1928 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 30 T + 531 T^{2} + 5948 T^{3} + 531 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 9 T + 222 T^{2} - 1608 T^{3} + 222 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.584447899011735667339580925616, −8.172579301285071387617011357481, −8.169610470247004261151053069330, −7.988367821801163022835933366548, −7.51702056163198319132966030292, −7.41956921838051250455418890683, −6.82377209018436714601766899280, −6.81313382867175619352484404742, −6.52497699449911203536753541489, −6.29817554689136358218635620268, −5.87338081832233059754996266254, −5.74828110519772520210281619439, −5.29990444401821595532093900436, −4.90408229360456541022233477896, −4.78767073186704803329905978589, −4.57077346435066343333511912803, −3.89303622839608625433553678016, −3.76723187475248474017651227626, −3.65522218727165515436571081491, −3.03818114620558777499039094258, −2.93381683845095450530871213637, −2.64700778070461331394599302020, −2.14756264929705504930733200813, −1.70809905523468441561561219361, −1.16317780908747699667837173688, 0, 0, 0,
1.16317780908747699667837173688, 1.70809905523468441561561219361, 2.14756264929705504930733200813, 2.64700778070461331394599302020, 2.93381683845095450530871213637, 3.03818114620558777499039094258, 3.65522218727165515436571081491, 3.76723187475248474017651227626, 3.89303622839608625433553678016, 4.57077346435066343333511912803, 4.78767073186704803329905978589, 4.90408229360456541022233477896, 5.29990444401821595532093900436, 5.74828110519772520210281619439, 5.87338081832233059754996266254, 6.29817554689136358218635620268, 6.52497699449911203536753541489, 6.81313382867175619352484404742, 6.82377209018436714601766899280, 7.41956921838051250455418890683, 7.51702056163198319132966030292, 7.988367821801163022835933366548, 8.169610470247004261151053069330, 8.172579301285071387617011357481, 8.584447899011735667339580925616
