| L(s) = 1 | − 3·3-s + 7-s − 3·9-s − 36·11-s − 5·13-s + 2·19-s − 3·21-s − 99·23-s + 18·27-s − 63·29-s − 7·31-s + 108·33-s + 64·37-s + 15·39-s − 18·41-s + 46·43-s + 81·47-s + 24·49-s − 6·57-s + 126·59-s + 41·61-s − 3·63-s − 116·67-s + 297·69-s − 86·73-s − 36·77-s + 83·79-s + ⋯ |
| L(s) = 1 | − 3-s + 1/7·7-s − 1/3·9-s − 3.27·11-s − 0.384·13-s + 2/19·19-s − 1/7·21-s − 4.30·23-s + 2/3·27-s − 2.17·29-s − 0.225·31-s + 3.27·33-s + 1.72·37-s + 5/13·39-s − 0.439·41-s + 1.06·43-s + 1.72·47-s + 0.489·49-s − 0.105·57-s + 2.13·59-s + 0.672·61-s − 0.0476·63-s − 1.73·67-s + 4.30·69-s − 1.17·73-s − 0.467·77-s + 1.05·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6836550598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6836550598\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T + 4 p T^{2} + p^{3} T^{3} + p^{4} T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 - T - 23 T^{2} + 74 T^{3} - 1874 T^{4} + 74 p^{2} T^{5} - 23 p^{4} T^{6} - p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 36 T + 683 T^{2} + 9036 T^{3} + 100632 T^{4} + 9036 p^{2} T^{5} + 683 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 5 T - 245 T^{2} - 340 T^{3} + 40114 T^{4} - 340 p^{2} T^{5} - 245 p^{4} T^{6} + 5 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 769 T^{2} + 298176 T^{4} - 769 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - T + 648 T^{2} - p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 99 T + 5117 T^{2} + 183150 T^{3} + 4870902 T^{4} + 183150 p^{2} T^{5} + 5117 p^{4} T^{6} + 99 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 63 T + 2123 T^{2} + 50400 T^{3} + 1045362 T^{4} + 50400 p^{2} T^{5} + 2123 p^{4} T^{6} + 63 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 7 T - 1217 T^{2} - 4592 T^{3} + 632146 T^{4} - 4592 p^{2} T^{5} - 1217 p^{4} T^{6} + 7 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 32 T + 1806 T^{2} - 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T + 1913 T^{2} + 32490 T^{3} + 613812 T^{4} + 32490 p^{2} T^{5} + 1913 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 23 T - 1320 T^{2} - 23 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 81 T + 6929 T^{2} - 384102 T^{3} + 22437966 T^{4} - 384102 p^{2} T^{5} + 6929 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 7204 T^{2} + 26018214 T^{4} - 7204 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 126 T + 11993 T^{2} - 844326 T^{3} + 51207492 T^{4} - 844326 p^{2} T^{5} + 11993 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 41 T - 5513 T^{2} + 10168 T^{3} + 31652794 T^{4} + 10168 p^{2} T^{5} - 5513 p^{4} T^{6} - 41 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 116 T + 3787 T^{2} + 80156 T^{3} + 12934456 T^{4} + 80156 p^{2} T^{5} + 3787 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 18616 T^{2} + 137194926 T^{4} - 18616 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 43 T + 10452 T^{2} + 43 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 83 T - 5459 T^{2} + 11122 T^{3} + 70528774 T^{4} + 11122 p^{2} T^{5} - 5459 p^{4} T^{6} - 83 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 81 T + 16289 T^{2} - 1142262 T^{3} + 166474326 T^{4} - 1142262 p^{2} T^{5} + 16289 p^{4} T^{6} - 81 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6916 T^{2} + 69013446 T^{4} - 6916 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 196 T + 10291 T^{2} - 1824172 T^{3} + 341030200 T^{4} - 1824172 p^{2} T^{5} + 10291 p^{4} T^{6} - 196 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−7.31835141046983113698647050252, −6.81080866244786844555276225285, −6.38543633595567523196170768241, −6.11209422984367825251434061309, −5.99909673270763687510193106103, −5.86492925260912869715256433276, −5.69836838918942213213806104140, −5.67033742070856181659192533363, −5.09232809143428325646495516808, −5.08174640142163742058154001922, −4.80590955402051147454406111148, −4.54884904992508427978957580617, −4.22063280264743232849194901700, −3.91646515046829831885459645328, −3.68469995377549707094044855382, −3.45964381336728225859643362485, −3.09819098808163131659161543740, −2.59724446212499189219671257089, −2.31279629022733969251611824096, −2.25230433450886353565907901328, −2.07203652928047182032131861116, −1.69706340146706483474346770082, −0.68688850377529472329510687916, −0.55839292517058470997581046246, −0.25393457971646615224799112909,
0.25393457971646615224799112909, 0.55839292517058470997581046246, 0.68688850377529472329510687916, 1.69706340146706483474346770082, 2.07203652928047182032131861116, 2.25230433450886353565907901328, 2.31279629022733969251611824096, 2.59724446212499189219671257089, 3.09819098808163131659161543740, 3.45964381336728225859643362485, 3.68469995377549707094044855382, 3.91646515046829831885459645328, 4.22063280264743232849194901700, 4.54884904992508427978957580617, 4.80590955402051147454406111148, 5.08174640142163742058154001922, 5.09232809143428325646495516808, 5.67033742070856181659192533363, 5.69836838918942213213806104140, 5.86492925260912869715256433276, 5.99909673270763687510193106103, 6.11209422984367825251434061309, 6.38543633595567523196170768241, 6.81080866244786844555276225285, 7.31835141046983113698647050252
