| L(s) = 1 | + 4-s − 8·7-s + 4·16-s − 12·19-s + 10·25-s − 8·28-s − 6·31-s − 10·37-s − 28·43-s + 34·49-s + 30·61-s + 11·64-s + 2·67-s + 24·73-s − 12·76-s − 22·79-s + 10·100-s − 18·103-s + 2·109-s − 32·112-s − 2·121-s − 6·124-s + 127-s + 131-s + 96·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 3.02·7-s + 16-s − 2.75·19-s + 2·25-s − 1.51·28-s − 1.07·31-s − 1.64·37-s − 4.26·43-s + 34/7·49-s + 3.84·61-s + 11/8·64-s + 0.244·67-s + 2.80·73-s − 1.37·76-s − 2.47·79-s + 100-s − 1.77·103-s + 0.191·109-s − 3.02·112-s − 0.181·121-s − 0.538·124-s + 0.0887·127-s + 0.0873·131-s + 8.32·133-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6710206502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6710206502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^3$ | \( 1 + 2 T^{2} - 117 T^{4} + 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^3$ | \( 1 + 26 T^{2} + 387 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 26 T^{2} + 147 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 5 T - 12 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 34 T^{2} - 1053 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 86 T^{2} + 4587 T^{4} + 86 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 58 T^{2} - 117 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 - T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 12 T + 121 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 + 62 T^{2} - 4077 T^{4} + 62 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 191 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−9.307977702143290251685248366816, −8.877375778088524768538455432004, −8.692085276231817507210115235769, −8.389484416080281716421798073986, −8.272125857405220080718842926905, −8.070332322474198534288691873495, −7.42042119261756214635580008847, −6.94011348336572279146665150293, −6.85556458445052996666685167834, −6.72564319955406295052187638662, −6.52165438393630029415990490241, −6.49263226648858115597842504642, −5.93913334359049950948616290113, −5.32978516995454780332911919535, −5.31555823181706161529967955705, −5.24364929224346096197510818234, −4.26865055151552772092593418104, −4.14295084110367576915172927245, −3.75349606851646516654205374128, −3.27985420042283977712604627975, −3.13133603822800027139059085270, −2.89702056184428518777788709523, −2.01360747775334653379924745776, −1.94501038276519169271580926239, −0.50477077918082115811405497219,
0.50477077918082115811405497219, 1.94501038276519169271580926239, 2.01360747775334653379924745776, 2.89702056184428518777788709523, 3.13133603822800027139059085270, 3.27985420042283977712604627975, 3.75349606851646516654205374128, 4.14295084110367576915172927245, 4.26865055151552772092593418104, 5.24364929224346096197510818234, 5.31555823181706161529967955705, 5.32978516995454780332911919535, 5.93913334359049950948616290113, 6.49263226648858115597842504642, 6.52165438393630029415990490241, 6.72564319955406295052187638662, 6.85556458445052996666685167834, 6.94011348336572279146665150293, 7.42042119261756214635580008847, 8.070332322474198534288691873495, 8.272125857405220080718842926905, 8.389484416080281716421798073986, 8.692085276231817507210115235769, 8.877375778088524768538455432004, 9.307977702143290251685248366816
