| L(s) = 1 | + 2-s + 3·3-s + 8·4-s − 12·5-s + 3·6-s + 23·8-s − 12·10-s − 20·11-s + 24·12-s − 168·13-s − 36·15-s + 23·16-s + 96·17-s − 12·19-s − 96·20-s − 20·22-s + 176·23-s + 69·24-s + 125·25-s − 168·26-s − 27·27-s + 116·29-s − 36·30-s + 264·31-s + 184·32-s − 60·33-s + 96·34-s + ⋯ |
| L(s) = 1 | + 0.353·2-s + 0.577·3-s + 4-s − 1.07·5-s + 0.204·6-s + 1.01·8-s − 0.379·10-s − 0.548·11-s + 0.577·12-s − 3.58·13-s − 0.619·15-s + 0.359·16-s + 1.36·17-s − 0.144·19-s − 1.07·20-s − 0.193·22-s + 1.59·23-s + 0.586·24-s + 25-s − 1.26·26-s − 0.192·27-s + 0.742·29-s − 0.219·30-s + 1.52·31-s + 1.01·32-s − 0.316·33-s + 0.484·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21609 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.870654254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.870654254\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - 7 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 12 T + 19 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T - 931 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 84 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 96 T + 4303 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 12 T - 6715 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 176 T + 18809 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 264 T + 39905 T^{2} - 264 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 258 T + 15911 T^{2} + 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 156 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 408 T + 62641 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 722 T + 372407 T^{2} - 722 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 492 T + 36685 T^{2} + 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 492 T + 15083 T^{2} - 492 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 412 T - 131019 T^{2} + 412 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 296 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 240 T - 331417 T^{2} + 240 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 776 T + 109137 T^{2} + 776 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 924 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 744 T - 151433 T^{2} - 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 168 T + p^{3} T^{2} )^{2} \) |
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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
−12.60565661471013685762165510206, −12.24102599810251475786184219010, −11.98256732539835819681854399140, −11.58208353125720225941805798782, −10.73664593773720509217393393141, −10.28203702064239206441556067341, −10.03627259599089634942180402394, −9.225042833989514938367429944162, −8.585223767735285287400664909796, −7.64317755299277191550512389863, −7.63775813588671663128222601371, −7.19304235928931771188543906366, −6.68549046020235602212343745084, −5.49843539741683864808213251907, −4.78428184717168352257230697998, −4.64626434302029699052319027147, −3.43461730909433756698421576553, −2.58161775498685181372486855816, −2.45786968484432981470674604666, −0.74290047392436893700699512415,
0.74290047392436893700699512415, 2.45786968484432981470674604666, 2.58161775498685181372486855816, 3.43461730909433756698421576553, 4.64626434302029699052319027147, 4.78428184717168352257230697998, 5.49843539741683864808213251907, 6.68549046020235602212343745084, 7.19304235928931771188543906366, 7.63775813588671663128222601371, 7.64317755299277191550512389863, 8.585223767735285287400664909796, 9.225042833989514938367429944162, 10.03627259599089634942180402394, 10.28203702064239206441556067341, 10.73664593773720509217393393141, 11.58208353125720225941805798782, 11.98256732539835819681854399140, 12.24102599810251475786184219010, 12.60565661471013685762165510206
