| L(s) = 1 | − 2·4-s + 10·7-s + 4·16-s + 6·19-s + 4·25-s − 20·28-s − 18·31-s + 8·37-s − 28·43-s + 61·49-s − 18·61-s − 16·64-s − 4·67-s + 42·73-s − 12·76-s + 8·79-s − 8·100-s + 48·103-s − 34·109-s + 40·112-s + 10·121-s + 36·124-s + 127-s + 131-s + 60·133-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s + 3.77·7-s + 16-s + 1.37·19-s + 4/5·25-s − 3.77·28-s − 3.23·31-s + 1.31·37-s − 4.26·43-s + 61/7·49-s − 2.30·61-s − 2·64-s − 0.488·67-s + 4.91·73-s − 1.37·76-s + 0.900·79-s − 4/5·100-s + 4.72·103-s − 3.25·109-s + 3.77·112-s + 0.909·121-s + 3.23·124-s + 0.0887·127-s + 0.0873·131-s + 5.20·133-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
| \(L(1)\) |
\(\approx\) |
\(1.962349578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.962349578\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| good | 2 | $C_2$$\times$$C_2^2$ | \( ( 1 + p T^{2} )^{2}( 1 - p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^3$ | \( 1 - 4 T^{2} - 9 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 10 T^{2} - 21 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 28 T^{2} + 495 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 3 T + 22 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 4 T^{2} - 513 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 4 T - 21 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{4} \) |
| 47 | $C_2^3$ | \( 1 - 40 T^{2} - 609 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 + 104 T^{2} + 8007 T^{4} + 104 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 + 98 T^{2} + 6123 T^{4} + 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 21 T + 220 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 172 T^{2} + 21663 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 119 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−9.196388511497976356199062169067, −8.876555614053637804120198106225, −8.519379868362428858343814575190, −8.463681845534553039326776225981, −8.238439029753145795952760410200, −7.68868794836723936737761501634, −7.65998398712494571806786748335, −7.59746714390585447706212114441, −7.33341581612845252273168719148, −6.70539426307136703355588343480, −6.52859308162620839699364821736, −5.83485878016172540859065191283, −5.80396961269920167739436713063, −5.11314226612979465325326970094, −5.05064475941037451959452346539, −4.99544349149552491414040248634, −4.88792383534151970113164416805, −4.28653418143323724872235954554, −3.93012448432945952561673779543, −3.43660598060250442539838764466, −3.36655132950682273292083537907, −2.43187754419772588526089620704, −1.90748579708039254720588099870, −1.58810241130337706455791704347, −1.11851154836978925579362295626,
1.11851154836978925579362295626, 1.58810241130337706455791704347, 1.90748579708039254720588099870, 2.43187754419772588526089620704, 3.36655132950682273292083537907, 3.43660598060250442539838764466, 3.93012448432945952561673779543, 4.28653418143323724872235954554, 4.88792383534151970113164416805, 4.99544349149552491414040248634, 5.05064475941037451959452346539, 5.11314226612979465325326970094, 5.80396961269920167739436713063, 5.83485878016172540859065191283, 6.52859308162620839699364821736, 6.70539426307136703355588343480, 7.33341581612845252273168719148, 7.59746714390585447706212114441, 7.65998398712494571806786748335, 7.68868794836723936737761501634, 8.238439029753145795952760410200, 8.463681845534553039326776225981, 8.519379868362428858343814575190, 8.876555614053637804120198106225, 9.196388511497976356199062169067
