L(s) = 1 | − 3·2-s − 3-s + 6·4-s + 3·6-s − 4·7-s − 10·8-s − 9-s − 5·11-s − 6·12-s − 8·13-s + 12·14-s + 15·16-s + 7·17-s + 3·18-s − 19-s + 4·21-s + 15·22-s − 4·23-s + 10·24-s + 24·26-s − 2·27-s − 24·28-s − 14·29-s + 6·31-s − 21·32-s + 5·33-s − 21·34-s + ⋯ |
L(s) = 1 | − 2.12·2-s − 0.577·3-s + 3·4-s + 1.22·6-s − 1.51·7-s − 3.53·8-s − 1/3·9-s − 1.50·11-s − 1.73·12-s − 2.21·13-s + 3.20·14-s + 15/4·16-s + 1.69·17-s + 0.707·18-s − 0.229·19-s + 0.872·21-s + 3.19·22-s − 0.834·23-s + 2.04·24-s + 4.70·26-s − 0.384·27-s − 4.53·28-s − 2.59·29-s + 1.07·31-s − 3.71·32-s + 0.870·33-s − 3.60·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 5^{6} \cdot 37^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 3 | $S_4\times C_2$ | \( 1 + T + 2 T^{2} + 5 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 4 T + 19 T^{2} + 50 T^{3} + 19 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 5 T + 24 T^{2} + 59 T^{3} + 24 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 8 T + 53 T^{2} + 206 T^{3} + 53 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 T + 60 T^{2} - 235 T^{3} + 60 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + T + 32 T^{2} + 49 T^{3} + 32 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 4 T + 45 T^{2} + 136 T^{3} + 45 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 14 T + 123 T^{2} + 788 T^{3} + 123 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 6 T + 57 T^{2} - 158 T^{3} + 57 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 19 T + 174 T^{2} - 1159 T^{3} + 174 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 16 T + 133 T^{2} + 812 T^{3} + 133 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 53 | $S_4\times C_2$ | \( 1 - 6 T + 75 T^{2} - 420 T^{3} + 75 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 8 T + 117 T^{2} + 1028 T^{3} + 117 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 12 T + 165 T^{2} - 1226 T^{3} + 165 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 3 T + 126 T^{2} + 419 T^{3} + 126 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 8 T + 159 T^{2} - 758 T^{3} + 159 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 13 T + 250 T^{2} + 1889 T^{3} + 250 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 2 T + 137 T^{2} - 404 T^{3} + 137 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 19 T + 324 T^{2} - 3217 T^{3} + 324 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - T + 78 T^{2} - 31 T^{3} + 78 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 14 T + 287 T^{2} + 2708 T^{3} + 287 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−8.734671166136427204870632918819, −8.173182435395257165521158046602, −7.994495575888687250059781649198, −7.77248253254193019822401590837, −7.74601091024062655205762136804, −7.34180286149514231766113099271, −7.24700021762253524862626456436, −6.83465208904565015238105270550, −6.54505990865091987913140605835, −6.36033259953608977077414691603, −5.80446653765215366337044379923, −5.78132101808300318308149730057, −5.62578311779360009185543949590, −5.16424833681373195733162844713, −4.88423769562506654079926811634, −4.66867185435946907554061808349, −3.83630945070815497250331417604, −3.56668292369494028811310697636, −3.53808184843350127664256815147, −2.79500626875052348091352224436, −2.70795732868845213274132102179, −2.31772836902885995639686810395, −2.17219856844586864990858923792, −1.46839150036347553571859932344, −1.00378992318971350539312614864, 0, 0, 0,
1.00378992318971350539312614864, 1.46839150036347553571859932344, 2.17219856844586864990858923792, 2.31772836902885995639686810395, 2.70795732868845213274132102179, 2.79500626875052348091352224436, 3.53808184843350127664256815147, 3.56668292369494028811310697636, 3.83630945070815497250331417604, 4.66867185435946907554061808349, 4.88423769562506654079926811634, 5.16424833681373195733162844713, 5.62578311779360009185543949590, 5.78132101808300318308149730057, 5.80446653765215366337044379923, 6.36033259953608977077414691603, 6.54505990865091987913140605835, 6.83465208904565015238105270550, 7.24700021762253524862626456436, 7.34180286149514231766113099271, 7.74601091024062655205762136804, 7.77248253254193019822401590837, 7.994495575888687250059781649198, 8.173182435395257165521158046602, 8.734671166136427204870632918819