| L(s) = 1 | + 2·2-s − 4-s + 3·7-s − 5·8-s − 2·9-s − 7·11-s − 5·13-s + 6·14-s − 16-s − 5·17-s − 4·18-s − 6·19-s − 14·22-s + 3·23-s − 10·26-s + 7·27-s − 3·28-s − 12·29-s − 4·31-s + 4·32-s − 10·34-s + 2·36-s − 3·37-s − 12·38-s + 41-s − 2·43-s + 7·44-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1/2·4-s + 1.13·7-s − 1.76·8-s − 2/3·9-s − 2.11·11-s − 1.38·13-s + 1.60·14-s − 1/4·16-s − 1.21·17-s − 0.942·18-s − 1.37·19-s − 2.98·22-s + 0.625·23-s − 1.96·26-s + 1.34·27-s − 0.566·28-s − 2.22·29-s − 0.718·31-s + 0.707·32-s − 1.71·34-s + 1/3·36-s − 0.493·37-s − 1.94·38-s + 0.156·41-s − 0.304·43-s + 1.05·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{3} \) |
| 23 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 2 | $A_4\times C_2$ | \( 1 - p T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \) |
| 3 | $A_4\times C_2$ | \( 1 + 2 T^{2} - 7 T^{3} + 2 p T^{4} + p^{3} T^{6} \) |
| 11 | $A_4\times C_2$ | \( 1 + 7 T + 47 T^{2} + 161 T^{3} + 47 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $A_4\times C_2$ | \( 1 + 5 T + 17 T^{2} + 33 T^{3} + 17 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $A_4\times C_2$ | \( 1 + 5 T + 43 T^{2} + 129 T^{3} + 43 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $A_4\times C_2$ | \( 1 + 6 T + 62 T^{2} + 215 T^{3} + 62 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $A_4\times C_2$ | \( 1 + 12 T + 107 T^{2} + 592 T^{3} + 107 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 31 | $A_4\times C_2$ | \( 1 + 4 T + 96 T^{2} + 247 T^{3} + 96 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $A_4\times C_2$ | \( 1 + 3 T + 93 T^{2} + 195 T^{3} + 93 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $A_4\times C_2$ | \( 1 - T + 58 T^{2} + 87 T^{3} + 58 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $A_4\times C_2$ | \( 1 + 2 T + 128 T^{2} + 171 T^{3} + 128 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $A_4\times C_2$ | \( 1 + 12 T + 140 T^{2} + 1087 T^{3} + 140 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $A_4\times C_2$ | \( 1 - 9 T + 11 T^{2} + 419 T^{3} + 11 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $A_4\times C_2$ | \( 1 + 21 T + 303 T^{2} + 2681 T^{3} + 303 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $A_4\times C_2$ | \( 1 + 9 T + 161 T^{2} + 887 T^{3} + 161 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $A_4\times C_2$ | \( 1 + 13 T + 185 T^{2} + 1365 T^{3} + 185 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $A_4\times C_2$ | \( 1 + 16 T + 198 T^{2} + 1809 T^{3} + 198 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $A_4\times C_2$ | \( 1 - 9 T + 197 T^{2} - 1285 T^{3} + 197 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $A_4\times C_2$ | \( 1 + 18 T + 282 T^{2} + 2493 T^{3} + 282 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $A_4\times C_2$ | \( 1 - 29 T + 443 T^{2} - 4603 T^{3} + 443 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $A_4\times C_2$ | \( 1 - 7 T + 253 T^{2} - 1155 T^{3} + 253 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $A_4\times C_2$ | \( 1 + T + 142 T^{2} + 277 T^{3} + 142 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.68505204303610114232881314017, −7.60106236153447593219611580103, −7.46006098675106934615307436057, −7.17314791331029379251449891841, −6.88624394833029820502249280571, −6.53052063550509640138981441180, −6.16720011801809404395514743597, −5.81300856214236181626568139921, −5.73512951546511089849341820618, −5.62534245099284999875883621989, −5.07975734053058104781991530741, −4.97652169514797110214784827238, −4.68638764437823261678484854732, −4.63726552858375848837168701296, −4.52288105134611577803676430120, −4.33074518946051562218341403003, −3.53650762062777632586206897457, −3.47411548810944810959236266043, −3.36997547183104508126559043005, −2.82685451019283803031319590873, −2.55375512934309413234165045729, −2.23753790580279990773571128292, −1.92346582638956421991183983702, −1.72549539683691036272509663868, −1.00534766521534650807430766966, 0, 0, 0,
1.00534766521534650807430766966, 1.72549539683691036272509663868, 1.92346582638956421991183983702, 2.23753790580279990773571128292, 2.55375512934309413234165045729, 2.82685451019283803031319590873, 3.36997547183104508126559043005, 3.47411548810944810959236266043, 3.53650762062777632586206897457, 4.33074518946051562218341403003, 4.52288105134611577803676430120, 4.63726552858375848837168701296, 4.68638764437823261678484854732, 4.97652169514797110214784827238, 5.07975734053058104781991530741, 5.62534245099284999875883621989, 5.73512951546511089849341820618, 5.81300856214236181626568139921, 6.16720011801809404395514743597, 6.53052063550509640138981441180, 6.88624394833029820502249280571, 7.17314791331029379251449891841, 7.46006098675106934615307436057, 7.60106236153447593219611580103, 7.68505204303610114232881314017
