| L(s) = 1 | + 3·3-s − 4·5-s − 4·7-s + 6·11-s + 4·13-s − 12·15-s + 9·17-s + 5·19-s − 12·21-s + 2·23-s + 10·25-s − 10·27-s − 3·29-s + 31-s + 18·33-s + 16·35-s − 11·37-s + 12·39-s − 5·41-s + 12·43-s + 4·47-s + 10·49-s + 27·51-s + 12·53-s − 24·55-s + 15·57-s + 11·59-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 1.78·5-s − 1.51·7-s + 1.80·11-s + 1.10·13-s − 3.09·15-s + 2.18·17-s + 1.14·19-s − 2.61·21-s + 0.417·23-s + 2·25-s − 1.92·27-s − 0.557·29-s + 0.179·31-s + 3.13·33-s + 2.70·35-s − 1.80·37-s + 1.92·39-s − 0.780·41-s + 1.82·43-s + 0.583·47-s + 10/7·49-s + 3.78·51-s + 1.64·53-s − 3.23·55-s + 1.98·57-s + 1.43·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.66581803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.66581803\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^3: C_4$ | \( 1 - p T + p^{2} T^{2} - 17 T^{3} + 32 T^{4} - 17 p T^{5} + p^{4} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 6 T + 32 T^{2} - 118 T^{3} + 398 T^{4} - 118 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 9 T + 75 T^{2} - 427 T^{3} + 1988 T^{4} - 427 p T^{5} + 75 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 5 T + 79 T^{2} - 279 T^{3} + 2276 T^{4} - 279 p T^{5} + 79 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 2 T + 68 T^{2} - 130 T^{3} + 2086 T^{4} - 130 p T^{5} + 68 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 3 T + 79 T^{2} + 285 T^{3} + 2896 T^{4} + 285 p T^{5} + 79 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $((C_8 : C_2):C_2):C_2$ | \( 1 - T + 101 T^{2} - 75 T^{3} + 4392 T^{4} - 75 p T^{5} + 101 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 11 T + 119 T^{2} + 757 T^{3} + 5392 T^{4} + 757 p T^{5} + 119 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 5 T + 31 T^{2} - 377 T^{3} - 1844 T^{4} - 377 p T^{5} + 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 12 T + 192 T^{2} - 1452 T^{3} + 12606 T^{4} - 1452 p T^{5} + 192 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 2 T + 78 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 11 T + 173 T^{2} - 1653 T^{3} + 13184 T^{4} - 1653 p T^{5} + 173 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 19 T + 329 T^{2} - 3879 T^{3} + 35124 T^{4} - 3879 p T^{5} + 329 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $((C_8 : C_2):C_2):C_2$ | \( 1 + 8 T - 32 T^{2} - 440 T^{3} + 622 T^{4} - 440 p T^{5} - 32 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 248 T^{2} - 1512 T^{3} + 26382 T^{4} - 1512 p T^{5} + 248 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 13 T + 169 T^{2} - 969 T^{3} + 10124 T^{4} - 969 p T^{5} + 169 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 8 T + 152 T^{2} - 1480 T^{3} + 12286 T^{4} - 1480 p T^{5} + 152 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 25 T + 363 T^{2} - 4667 T^{3} + 51780 T^{4} - 4667 p T^{5} + 363 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $((C_8 : C_2):C_2):C_2$ | \( 1 - 18 T + 212 T^{2} - 222 T^{3} - 2186 T^{4} - 222 p T^{5} + 212 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.11421904193602056197510330097, −5.77817203900785957015262716269, −5.65967566902144361904691826623, −5.49716228762688941833075681116, −5.38697606373562795889220758544, −4.88788294310038060175131097773, −4.82759722654380059826067951943, −4.55574463889217641490117909865, −4.34297555559043945540694254356, −3.78852367788728738351178154260, −3.72611943405535145678491323419, −3.68489666453329605537091603861, −3.65512013985341843805823490562, −3.37498204257911760800811433131, −3.17158449475702291384005354620, −3.06649679066555061899545824157, −3.02112980269086870654108432361, −2.24864501631302080887951834213, −2.21427354521559710151736097156, −2.17368016616571420951213944137, −1.67840814059649002135743347666, −1.11444452364011462504642501112, −0.805737653323568314788110739845, −0.68162738238628253825191714430, −0.61835439471621721868960779322,
0.61835439471621721868960779322, 0.68162738238628253825191714430, 0.805737653323568314788110739845, 1.11444452364011462504642501112, 1.67840814059649002135743347666, 2.17368016616571420951213944137, 2.21427354521559710151736097156, 2.24864501631302080887951834213, 3.02112980269086870654108432361, 3.06649679066555061899545824157, 3.17158449475702291384005354620, 3.37498204257911760800811433131, 3.65512013985341843805823490562, 3.68489666453329605537091603861, 3.72611943405535145678491323419, 3.78852367788728738351178154260, 4.34297555559043945540694254356, 4.55574463889217641490117909865, 4.82759722654380059826067951943, 4.88788294310038060175131097773, 5.38697606373562795889220758544, 5.49716228762688941833075681116, 5.65967566902144361904691826623, 5.77817203900785957015262716269, 6.11421904193602056197510330097
