| L(s) = 1 | + 2·2-s + 4-s + 7-s − 2·8-s − 3·11-s − 2·13-s + 2·14-s − 4·16-s − 18·17-s + 2·19-s − 6·22-s + 3·23-s − 4·26-s + 28-s − 3·29-s + 2·31-s − 2·32-s − 36·34-s + 16·37-s + 4·38-s − 6·41-s − 17·43-s − 3·44-s + 6·46-s + 9·47-s + 6·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.377·7-s − 0.707·8-s − 0.904·11-s − 0.554·13-s + 0.534·14-s − 16-s − 4.36·17-s + 0.458·19-s − 1.27·22-s + 0.625·23-s − 0.784·26-s + 0.188·28-s − 0.557·29-s + 0.359·31-s − 0.353·32-s − 6.17·34-s + 2.63·37-s + 0.648·38-s − 0.937·41-s − 2.59·43-s − 0.452·44-s + 0.884·46-s + 1.31·47-s + 6/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.080155599\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.080155599\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T - 5 T^{2} + 8 T^{3} - 20 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 3 T - 7 T^{2} - 18 T^{3} + 36 T^{4} - 18 p T^{5} - 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 2 T + 10 T^{2} - 64 T^{3} - 185 T^{4} - 64 p T^{5} + 10 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 9 T + 46 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 3 T - 31 T^{2} + 18 T^{3} + 864 T^{4} + 18 p T^{5} - 31 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3 T - 43 T^{2} - 18 T^{3} + 1602 T^{4} - 18 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 2 T - 26 T^{2} + 64 T^{3} - 185 T^{4} + 64 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 17 T + 139 T^{2} + 1088 T^{3} + 8224 T^{4} + 1088 p T^{5} + 139 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9 T - 25 T^{2} - 108 T^{3} + 5220 T^{4} - 108 p T^{5} - 25 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 3 T - 103 T^{2} + 18 T^{3} + 8532 T^{4} + 18 p T^{5} - 103 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + T - 47 T^{2} - 74 T^{3} - 1478 T^{4} - 74 p T^{5} - 47 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2$ | \( ( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 11 T + 102 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 2 T - 75 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 9 T - 97 T^{2} + 108 T^{3} + 18072 T^{4} + 108 p T^{5} - 97 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 15 T + 160 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 11 T - 95 T^{2} + 242 T^{3} + 25510 T^{4} + 242 p T^{5} - 95 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.85939501111415035919907215913, −6.64360564819188993231464793411, −6.37221030211350448392634298960, −6.36858332515990500959934303678, −5.82101939346536012136074021965, −5.58035663198225969691658068112, −5.43063259331106849398368286666, −5.38505988644591281667851504292, −5.05287122772067412600783394698, −4.77247838521949474450934730247, −4.49309436647894845531686685883, −4.37766815554762989460098229166, −4.28606288454581634274843902220, −4.16250271313971930805068668779, −3.66027291201539329773769919083, −3.52963761548160609499957729765, −2.94958009748604298446120992469, −2.89505671910574810782854984780, −2.57623469464808744662191450288, −2.47099230915486505267427229671, −1.96516205169217768083173587473, −1.91976876368333965946803393861, −1.42986992319072475202883683175, −0.57768095327140122740756497340, −0.39325568901143939673284207071,
0.39325568901143939673284207071, 0.57768095327140122740756497340, 1.42986992319072475202883683175, 1.91976876368333965946803393861, 1.96516205169217768083173587473, 2.47099230915486505267427229671, 2.57623469464808744662191450288, 2.89505671910574810782854984780, 2.94958009748604298446120992469, 3.52963761548160609499957729765, 3.66027291201539329773769919083, 4.16250271313971930805068668779, 4.28606288454581634274843902220, 4.37766815554762989460098229166, 4.49309436647894845531686685883, 4.77247838521949474450934730247, 5.05287122772067412600783394698, 5.38505988644591281667851504292, 5.43063259331106849398368286666, 5.58035663198225969691658068112, 5.82101939346536012136074021965, 6.36858332515990500959934303678, 6.37221030211350448392634298960, 6.64360564819188993231464793411, 6.85939501111415035919907215913
