| L(s) = 1 | + 4·2-s − 6·3-s + 10·4-s + 2·5-s − 24·6-s − 2·7-s + 20·8-s + 21·9-s + 8·10-s − 60·12-s − 4·13-s − 8·14-s − 12·15-s + 35·16-s + 84·18-s − 4·19-s + 20·20-s + 12·21-s − 120·24-s + 25-s − 16·26-s − 54·27-s − 20·28-s + 6·29-s − 48·30-s + 8·31-s + 56·32-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 3.46·3-s + 5·4-s + 0.894·5-s − 9.79·6-s − 0.755·7-s + 7.07·8-s + 7·9-s + 2.52·10-s − 17.3·12-s − 1.10·13-s − 2.13·14-s − 3.09·15-s + 35/4·16-s + 19.7·18-s − 0.917·19-s + 4.47·20-s + 2.61·21-s − 24.4·24-s + 1/5·25-s − 3.13·26-s − 10.3·27-s − 3.77·28-s + 1.11·29-s − 8.76·30-s + 1.43·31-s + 9.89·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.061952489\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.061952489\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_2^3$ | \( 1 - 4 T^{2} - 105 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 4 T - 8 T^{2} - 56 T^{3} - 89 T^{4} - 56 p T^{5} - 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6 T - 13 T^{2} + 54 T^{3} + 516 T^{4} + 54 p T^{5} - 13 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 4 T + 48 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 8 T - 8 T^{2} + 16 T^{3} + 1447 T^{4} + 16 p T^{5} - 8 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 85 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 20 T^{2} + 216 T^{3} + 4935 T^{4} + 216 p T^{5} + 20 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 136 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 26 T^{2} + 448 T^{3} - 1901 T^{4} + 448 p T^{5} - 26 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T - 110 T^{2} - 272 T^{3} + 4915 T^{4} - 272 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.35452537391707474234089044469, −6.78837219976460915282642804045, −6.77999265447686058270460046657, −6.74636387326027972868716480144, −6.55867219498931389760065422526, −6.37403069628029111972849825689, −5.99491434951561075022110030736, −5.96824082078872918442288110566, −5.74066841585952262322010629659, −5.41222437166460181493742530735, −5.04683163232396621109634286138, −4.99944674627010406895746814650, −4.93343564920020681877971557824, −4.51941725889299800964792239875, −4.48972574018327275532394879511, −4.15859256428984410742948354755, −3.60784697836687808586014191675, −3.52766115861853087528864958122, −3.19472072902014304202356822692, −2.57737460756336344666404610470, −2.32563700120707682699322039336, −2.16911008402207231928102689453, −1.42410851180371216897370782545, −1.23657382790840338906751105788, −0.45156741339020191454322610194,
0.45156741339020191454322610194, 1.23657382790840338906751105788, 1.42410851180371216897370782545, 2.16911008402207231928102689453, 2.32563700120707682699322039336, 2.57737460756336344666404610470, 3.19472072902014304202356822692, 3.52766115861853087528864958122, 3.60784697836687808586014191675, 4.15859256428984410742948354755, 4.48972574018327275532394879511, 4.51941725889299800964792239875, 4.93343564920020681877971557824, 4.99944674627010406895746814650, 5.04683163232396621109634286138, 5.41222437166460181493742530735, 5.74066841585952262322010629659, 5.96824082078872918442288110566, 5.99491434951561075022110030736, 6.37403069628029111972849825689, 6.55867219498931389760065422526, 6.74636387326027972868716480144, 6.77999265447686058270460046657, 6.78837219976460915282642804045, 7.35452537391707474234089044469
