| L(s) = 1 | − 6·3-s − 6·5-s + 2·7-s + 21·9-s − 6·11-s + 36·15-s − 12·21-s + 17·25-s − 54·27-s + 36·29-s + 30·31-s + 36·33-s − 12·35-s − 126·45-s + 7·49-s + 6·53-s + 36·55-s + 18·59-s + 42·63-s − 102·75-s − 12·77-s − 10·79-s + 108·81-s − 216·87-s − 180·93-s − 126·99-s + 12·101-s + ⋯ |
| L(s) = 1 | − 3.46·3-s − 2.68·5-s + 0.755·7-s + 7·9-s − 1.80·11-s + 9.29·15-s − 2.61·21-s + 17/5·25-s − 10.3·27-s + 6.68·29-s + 5.38·31-s + 6.26·33-s − 2.02·35-s − 18.7·45-s + 49-s + 0.824·53-s + 4.85·55-s + 2.34·59-s + 5.29·63-s − 11.7·75-s − 1.36·77-s − 1.12·79-s + 12·81-s − 23.1·87-s − 18.6·93-s − 12.6·99-s + 1.19·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5925530762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5925530762\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| good | 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 18 T + 137 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 69 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 37 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 94 T^{2} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 167 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
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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
−7.66585194998326968733599280914, −7.13256520472491480462266831997, −6.93886477268401285725720758426, −6.82506143933126905702958236818, −6.64211017116379267075769928475, −6.45089987373417682913018992291, −6.04744429158877863800721790269, −5.95013945145638840028639771196, −5.82976806317769829452527945115, −5.04525844219978013053040846727, −4.98481475203830793227457046205, −4.89186855058007423822224606451, −4.83449146990163910538163488310, −4.59610847647984613228342281586, −4.23327953835424269660308054665, −4.08274524128991227999016179734, −3.94396487484637953235084974846, −3.13159524331979621957906709923, −2.86650789734269046157605535248, −2.67877329282256188867752941831, −2.46617499730207489343222664795, −1.38505331364018386573061687479, −0.808786369348333386255620335300, −0.72558354174778886764003036679, −0.67348201595169164132173305672,
0.67348201595169164132173305672, 0.72558354174778886764003036679, 0.808786369348333386255620335300, 1.38505331364018386573061687479, 2.46617499730207489343222664795, 2.67877329282256188867752941831, 2.86650789734269046157605535248, 3.13159524331979621957906709923, 3.94396487484637953235084974846, 4.08274524128991227999016179734, 4.23327953835424269660308054665, 4.59610847647984613228342281586, 4.83449146990163910538163488310, 4.89186855058007423822224606451, 4.98481475203830793227457046205, 5.04525844219978013053040846727, 5.82976806317769829452527945115, 5.95013945145638840028639771196, 6.04744429158877863800721790269, 6.45089987373417682913018992291, 6.64211017116379267075769928475, 6.82506143933126905702958236818, 6.93886477268401285725720758426, 7.13256520472491480462266831997, 7.66585194998326968733599280914
