| L(s) = 1 | − 1.04e3·7-s + 688·13-s + 4.64e3·19-s + 2.72e4·25-s + 2.11e4·31-s − 4.81e4·37-s + 1.81e5·43-s + 5.88e5·49-s + 5.02e5·61-s + 4.32e5·67-s − 6.16e5·73-s + 1.08e6·79-s − 7.21e5·91-s − 7.43e4·97-s − 2.92e6·103-s − 2.87e6·109-s + 2.79e6·121-s + 127-s + 131-s − 4.86e6·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 3.05·7-s + 0.313·13-s + 0.676·19-s + 1.74·25-s + 0.709·31-s − 0.950·37-s + 2.28·43-s + 5.00·49-s + 2.21·61-s + 1.43·67-s − 1.58·73-s + 2.19·79-s − 0.956·91-s − 0.0814·97-s − 2.67·103-s − 2.21·109-s + 1.57·121-s − 2.06·133-s − 1.92·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.479912652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.479912652\) |
| \(L(4)\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 1088 p^{2} T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 524 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 2794034 T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 344 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9376 p^{2} T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2320 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 262974530 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 653627360 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 10564 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 24082 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2345166880 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 90952 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 4927927970 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5643898400 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82775997074 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 251138 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 216088 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 253293689090 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 308176 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 540124 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 215387844910 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 944049116864 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 37168 T + p^{6} T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.38619384200964713163541735196, −11.94819219920316582682917218961, −11.06285518294771001089617440636, −10.64023916214317475943296051644, −9.970624878099726940863214117185, −9.788215091197648744194926163940, −9.091176547676685476769026020454, −8.908636293553471719664983293792, −8.024120971737906136729213255414, −7.19970216617931207376401411647, −6.65366879608939218366303205123, −6.54712463077993813366066036242, −5.74410181526067886608122137324, −5.20588695502863101123929630959, −4.07601321271560429900367631506, −3.58280034854143074365999037839, −2.92515428495083839853314816637, −2.53893030570814876601858443627, −1.00809641632729382430677895798, −0.45119025551434722694204636363,
0.45119025551434722694204636363, 1.00809641632729382430677895798, 2.53893030570814876601858443627, 2.92515428495083839853314816637, 3.58280034854143074365999037839, 4.07601321271560429900367631506, 5.20588695502863101123929630959, 5.74410181526067886608122137324, 6.54712463077993813366066036242, 6.65366879608939218366303205123, 7.19970216617931207376401411647, 8.024120971737906136729213255414, 8.908636293553471719664983293792, 9.091176547676685476769026020454, 9.788215091197648744194926163940, 9.970624878099726940863214117185, 10.64023916214317475943296051644, 11.06285518294771001089617440636, 11.94819219920316582682917218961, 12.38619384200964713163541735196
