| L(s) = 1 | − 2-s + 3-s − 2·5-s − 6-s + 8-s + 2·10-s − 11-s − 2·13-s − 2·15-s − 16-s − 2·17-s + 22-s + 23-s + 24-s + 3·25-s + 2·26-s − 27-s − 29-s + 2·30-s − 2·31-s − 33-s + 2·34-s − 37-s − 2·39-s − 2·40-s − 43-s − 46-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s − 2·5-s − 6-s + 8-s + 2·10-s − 11-s − 2·13-s − 2·15-s − 16-s − 2·17-s + 22-s + 23-s + 24-s + 3·25-s + 2·26-s − 27-s − 29-s + 2·30-s − 2·31-s − 33-s + 2·34-s − 37-s − 2·39-s − 2·40-s − 43-s − 46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2433600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07271344533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07271344533\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.964457388979948521414627923794, −9.147289831313357227524566253403, −9.105839079062932772415921245324, −8.435799551363335534680430521302, −8.384728395520947154633446034378, −7.970039035930013137353741333474, −7.54165753383351798083423304148, −7.12727068054571383144478064430, −7.11357691749647592481615362864, −6.63267495309112503955734206570, −5.54522805494220525213457261892, −5.10603538624148301338216136181, −4.63580889277369440902474529441, −4.54981022853638669724911746761, −3.81313085642355852162802655686, −3.14258092897920719048371866679, −3.14042367874384205321215642021, −2.03260654616110431249341384994, −1.98377503334057917110767307884, −0.22320356149896026794745117078,
0.22320356149896026794745117078, 1.98377503334057917110767307884, 2.03260654616110431249341384994, 3.14042367874384205321215642021, 3.14258092897920719048371866679, 3.81313085642355852162802655686, 4.54981022853638669724911746761, 4.63580889277369440902474529441, 5.10603538624148301338216136181, 5.54522805494220525213457261892, 6.63267495309112503955734206570, 7.11357691749647592481615362864, 7.12727068054571383144478064430, 7.54165753383351798083423304148, 7.970039035930013137353741333474, 8.384728395520947154633446034378, 8.435799551363335534680430521302, 9.105839079062932772415921245324, 9.147289831313357227524566253403, 9.964457388979948521414627923794
