| L(s) = 1 | + 3·2-s + 3·3-s + 6·4-s + 9·6-s + 6·7-s + 10·8-s + 6·9-s + 8·11-s + 18·12-s + 10·13-s + 18·14-s + 15·16-s + 2·17-s + 18·18-s − 2·19-s + 18·21-s + 24·22-s + 2·23-s + 30·24-s + 30·26-s + 10·27-s + 36·28-s − 2·29-s − 3·31-s + 21·32-s + 24·33-s + 6·34-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 1.73·3-s + 3·4-s + 3.67·6-s + 2.26·7-s + 3.53·8-s + 2·9-s + 2.41·11-s + 5.19·12-s + 2.77·13-s + 4.81·14-s + 15/4·16-s + 0.485·17-s + 4.24·18-s − 0.458·19-s + 3.92·21-s + 5.11·22-s + 0.417·23-s + 6.12·24-s + 5.88·26-s + 1.92·27-s + 6.80·28-s − 0.371·29-s − 0.538·31-s + 3.71·32-s + 4.17·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{3} \cdot 5^{6} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(115.2469854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(115.2469854\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{3} \) |
| 5 | | \( 1 \) |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 11 | $S_4\times C_2$ | \( 1 - 8 T + 46 T^{2} - 168 T^{3} + 46 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 10 T + 64 T^{2} - 274 T^{3} + 64 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 2 T + 16 T^{2} - 72 T^{3} + 16 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 2 T + 22 T^{2} + 80 T^{3} + 22 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2 T + 37 T^{2} - 108 T^{3} + 37 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 2 T + 55 T^{2} + 132 T^{3} + 55 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 6 T + 47 T^{2} + 316 T^{3} + 47 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{3} \) |
| 43 | $S_4\times C_2$ | \( 1 + 53 T^{2} - 16 T^{3} + 53 p T^{4} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 2 T + 106 T^{2} + 192 T^{3} + 106 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 10 T + 115 T^{2} - 1068 T^{3} + 115 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 2 T + 101 T^{2} - 44 T^{3} + 101 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 14 T + 92 T^{2} - 336 T^{3} + 92 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 16 T + 186 T^{2} + 1492 T^{3} + 186 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 8 T + 134 T^{2} + 1156 T^{3} + 134 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2 T + 87 T^{2} - 100 T^{3} + 87 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 6 T + 230 T^{2} + 920 T^{3} + 230 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 2 T + 66 T^{2} - 576 T^{3} + 66 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 2 T + 123 T^{2} + 36 T^{3} + 123 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 20 T + 416 T^{2} + 4116 T^{3} + 416 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
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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
−7.38324460586880993160247519408, −6.97293386354697525757274537371, −6.80737085503567472327603019928, −6.70183141039366571111400964838, −6.23352035568258555413799655221, −6.04496532549210623255202117458, −6.02823160212645425067889299002, −5.34263937215714238639368577895, −5.33122774228024276704483599204, −5.25684490067060701954155744952, −4.51689152470067151918898524113, −4.37413274993218699327944851560, −4.37098813119203208882724021745, −3.92755062400044117387648243703, −3.90874648000684020053313416257, −3.66671662741372398099983284784, −3.16581043325352448648390500335, −3.01722486010701818453434503045, −3.00809419465626097115237118195, −2.00498685587403903589060274266, −1.97643656955873048076356377687, −1.94955308824885451806762104345, −1.21388883770716023775771094286, −1.21117304372157089683019747007, −1.19226655308512051834868637109,
1.19226655308512051834868637109, 1.21117304372157089683019747007, 1.21388883770716023775771094286, 1.94955308824885451806762104345, 1.97643656955873048076356377687, 2.00498685587403903589060274266, 3.00809419465626097115237118195, 3.01722486010701818453434503045, 3.16581043325352448648390500335, 3.66671662741372398099983284784, 3.90874648000684020053313416257, 3.92755062400044117387648243703, 4.37098813119203208882724021745, 4.37413274993218699327944851560, 4.51689152470067151918898524113, 5.25684490067060701954155744952, 5.33122774228024276704483599204, 5.34263937215714238639368577895, 6.02823160212645425067889299002, 6.04496532549210623255202117458, 6.23352035568258555413799655221, 6.70183141039366571111400964838, 6.80737085503567472327603019928, 6.97293386354697525757274537371, 7.38324460586880993160247519408
