| L(s) = 1 | + 12·4-s − 8·7-s − 8·9-s + 74·16-s − 96·28-s − 96·36-s + 16·37-s − 32·43-s + 28·49-s + 64·63-s + 300·64-s + 16·67-s + 16·79-s + 32·81-s − 16·109-s − 592·112-s + 64·121-s + 127-s + 131-s + 137-s + 139-s − 592·144-s + 192·148-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 6·4-s − 3.02·7-s − 8/3·9-s + 37/2·16-s − 18.1·28-s − 16·36-s + 2.63·37-s − 4.87·43-s + 4·49-s + 8.06·63-s + 75/2·64-s + 1.95·67-s + 1.80·79-s + 32/9·81-s − 1.53·109-s − 55.9·112-s + 5.81·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 49.3·144-s + 15.7·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 23^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.48609570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.48609570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | \( ( 1 + T^{2} )^{4} \) |
| good | 2 | \( ( 1 - 3 T^{2} + p^{2} T^{4} )^{4} \) |
| 5 | \( ( 1 + 48 T^{4} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 32 T^{2} + 466 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 - 36 T^{2} + 630 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 48 T^{2} + 1056 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 44 T^{2} + 1078 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 - 40 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - 56 T^{2} + 2608 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 4 T^{2} + 3238 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 136 T^{2} + 8944 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 188 T^{2} + 14326 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 + 136 T^{2} + 11008 T^{4} + 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 2640 T^{4} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 - 4 T - 24 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 188 T^{2} + 16870 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 212 T^{2} + 21862 T^{4} - 212 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 4 T - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 252 T^{2} + 28086 T^{4} + 252 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 176 T^{2} + 23424 T^{4} + 176 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 240 T^{2} + 28800 T^{4} - 240 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.91291058986125806844661571999, −4.81740687530280301544424816283, −4.60674312959301520381781649793, −4.39953434026058700569916691185, −4.18613275980708138312826889340, −3.89260520913756101580820335646, −3.75966258151502729681542939658, −3.56338865766601545154175217222, −3.31916120802819502118814422777, −3.26971908376332873195096957358, −3.26224931690181404946373666363, −3.19478368201143597473481859625, −2.96766282924794599627515445771, −2.84584193207384649665915629328, −2.66960821634312555231493263629, −2.60991650211926481388931996972, −2.41269204640216396390332163452, −2.22908336182441671361178267944, −1.99622964076765610773798920449, −1.86549875571546383250136321346, −1.82110370656645442699911043351, −1.58672464680082127656339911622, −1.14233022559293502845743716813, −0.57728473446826275768635673281, −0.52537393517743777731588283725,
0.52537393517743777731588283725, 0.57728473446826275768635673281, 1.14233022559293502845743716813, 1.58672464680082127656339911622, 1.82110370656645442699911043351, 1.86549875571546383250136321346, 1.99622964076765610773798920449, 2.22908336182441671361178267944, 2.41269204640216396390332163452, 2.60991650211926481388931996972, 2.66960821634312555231493263629, 2.84584193207384649665915629328, 2.96766282924794599627515445771, 3.19478368201143597473481859625, 3.26224931690181404946373666363, 3.26971908376332873195096957358, 3.31916120802819502118814422777, 3.56338865766601545154175217222, 3.75966258151502729681542939658, 3.89260520913756101580820335646, 4.18613275980708138312826889340, 4.39953434026058700569916691185, 4.60674312959301520381781649793, 4.81740687530280301544424816283, 4.91291058986125806844661571999
Plot not available for L-functions of degree greater than 10.
