| L(s) = 1 | − 4·3-s + 2·4-s − 8·7-s + 6·9-s − 8·12-s + 8·13-s − 16·19-s + 32·21-s + 16·25-s + 4·27-s − 16·28-s + 12·36-s − 32·39-s + 34·49-s + 16·52-s + 64·57-s − 40·61-s − 48·63-s − 8·64-s − 64·75-s − 32·76-s + 32·79-s − 37·81-s + 64·84-s − 64·91-s + 32·100-s + 8·108-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 4-s − 3.02·7-s + 2·9-s − 2.30·12-s + 2.21·13-s − 3.67·19-s + 6.98·21-s + 16/5·25-s + 0.769·27-s − 3.02·28-s + 2·36-s − 5.12·39-s + 34/7·49-s + 2.21·52-s + 8.47·57-s − 5.12·61-s − 6.04·63-s − 64-s − 7.39·75-s − 3.67·76-s + 3.60·79-s − 4.11·81-s + 6.98·84-s − 6.70·91-s + 16/5·100-s + 0.769·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2825860640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2825860640\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 44 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 74 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 140 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
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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
−9.225576258704113753876125665914, −9.146748128474840829353380223943, −9.132559562035687071910265120089, −8.496454584853849515614411450914, −8.391364747251461068586532976648, −8.242665093089288828162320377154, −7.51606245150690005120117675849, −7.10673328379375224030051821566, −6.76400738360322310421697647590, −6.72257724063785497298241506341, −6.45119263188054980877002830287, −6.20499814780678485213621427714, −6.19893415291904211512770343297, −5.98020452026363089909845191563, −5.66910146181115273914855359390, −4.95317988287184884838441410231, −4.82154067556283219880331755631, −4.32968932380572203867346331402, −4.07787527264030149165496335425, −3.35602937821452338064137109669, −3.24273203086242754636273006788, −2.82069919849367578603505346585, −2.34498505449897730852703937755, −1.41536962951545678532655654874, −0.44602461352603344099013286196,
0.44602461352603344099013286196, 1.41536962951545678532655654874, 2.34498505449897730852703937755, 2.82069919849367578603505346585, 3.24273203086242754636273006788, 3.35602937821452338064137109669, 4.07787527264030149165496335425, 4.32968932380572203867346331402, 4.82154067556283219880331755631, 4.95317988287184884838441410231, 5.66910146181115273914855359390, 5.98020452026363089909845191563, 6.19893415291904211512770343297, 6.20499814780678485213621427714, 6.45119263188054980877002830287, 6.72257724063785497298241506341, 6.76400738360322310421697647590, 7.10673328379375224030051821566, 7.51606245150690005120117675849, 8.242665093089288828162320377154, 8.391364747251461068586532976648, 8.496454584853849515614411450914, 9.132559562035687071910265120089, 9.146748128474840829353380223943, 9.225576258704113753876125665914
