| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 3·5-s + 4·6-s + 10·7-s − 2·8-s − 3·9-s + 6·10-s − 9·11-s + 2·12-s + 8·13-s + 20·14-s + 6·15-s − 4·16-s − 6·18-s − 12·19-s + 3·20-s + 20·21-s − 18·22-s + 3·23-s − 4·24-s + 5·25-s + 16·26-s − 14·27-s + 10·28-s + 12·30-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.34·5-s + 1.63·6-s + 3.77·7-s − 0.707·8-s − 9-s + 1.89·10-s − 2.71·11-s + 0.577·12-s + 2.21·13-s + 5.34·14-s + 1.54·15-s − 16-s − 1.41·18-s − 2.75·19-s + 0.670·20-s + 4.36·21-s − 3.83·22-s + 0.625·23-s − 0.816·24-s + 25-s + 3.13·26-s − 2.69·27-s + 1.88·28-s + 2.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.745638603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.745638603\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 9 T + 53 T^{2} + 234 T^{3} + 852 T^{4} + 234 p T^{5} + 53 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^3$ | \( 1 - 10 T^{2} - 189 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 9 T + 71 T^{2} - 396 T^{3} + 1812 T^{4} - 396 p T^{5} + 71 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 6 T - 10 T^{2} + 132 T^{3} - 441 T^{4} + 132 p T^{5} - 10 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 9 T + 94 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 49 T^{2} + 660 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 18 T + 218 T^{2} - 1980 T^{3} + 14967 T^{4} - 1980 p T^{5} + 218 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 3 T - 91 T^{2} - 18 T^{3} + 6714 T^{4} - 18 p T^{5} - 91 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 9 T + 17 T^{2} + 486 T^{3} - 4164 T^{4} + 486 p T^{5} + 17 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 27 T + 401 T^{2} + 4266 T^{3} + 36066 T^{4} + 4266 p T^{5} + 401 p^{2} T^{6} + 27 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 9 T + 143 T^{2} + 1044 T^{3} + 10776 T^{4} + 1044 p T^{5} + 143 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 208 T^{2} + 19710 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + T - 83 T^{2} - 74 T^{3} + 736 T^{4} - 74 p T^{5} - 83 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 190 T^{2} + 18051 T^{4} - 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 3 T - 163 T^{2} - 18 T^{3} + 20862 T^{4} - 18 p T^{5} - 163 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 13 T + 162 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−8.637912623621245661194937685205, −8.618265886824722937463030351511, −8.573822464289021449985997612454, −8.331434835036883978846217261003, −8.316735885740517866744288429769, −7.73908938220782076661718070503, −7.61027248529782336179099899338, −7.35946687067289968915583909942, −6.89368371965996315704867815918, −6.09197968479865225741494392262, −6.05733348575990498407424945560, −5.89491105206299944817048796683, −5.86133114823068535530816906042, −5.18533399889855185603035101661, −4.99777065149639036029178311490, −4.77850208837961334392755768453, −4.67813726596298679195548472200, −4.27363872271364831083112767078, −3.61659989837218548998120229511, −3.60954262628112269371643397405, −2.62218260627425433112634404589, −2.56362028717086246529103293973, −2.45851440435745749479068130137, −1.80394289706989757080037128275, −1.41862689644502559262185222382,
1.41862689644502559262185222382, 1.80394289706989757080037128275, 2.45851440435745749479068130137, 2.56362028717086246529103293973, 2.62218260627425433112634404589, 3.60954262628112269371643397405, 3.61659989837218548998120229511, 4.27363872271364831083112767078, 4.67813726596298679195548472200, 4.77850208837961334392755768453, 4.99777065149639036029178311490, 5.18533399889855185603035101661, 5.86133114823068535530816906042, 5.89491105206299944817048796683, 6.05733348575990498407424945560, 6.09197968479865225741494392262, 6.89368371965996315704867815918, 7.35946687067289968915583909942, 7.61027248529782336179099899338, 7.73908938220782076661718070503, 8.316735885740517866744288429769, 8.331434835036883978846217261003, 8.573822464289021449985997612454, 8.618265886824722937463030351511, 8.637912623621245661194937685205
