| L(s) = 1 | − 2-s + 4-s + 7-s − 2·8-s − 11-s + 13-s − 14-s + 2·16-s + 2·17-s + 22-s − 25-s − 26-s + 28-s + 29-s − 2·32-s − 2·34-s + 2·41-s − 44-s − 47-s + 49-s + 50-s + 52-s − 2·56-s − 58-s + 3·64-s + 67-s + 2·68-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s + 7-s − 2·8-s − 11-s + 13-s − 14-s + 2·16-s + 2·17-s + 22-s − 25-s − 26-s + 28-s + 29-s − 2·32-s − 2·34-s + 2·41-s − 44-s − 47-s + 49-s + 50-s + 52-s − 2·56-s − 58-s + 3·64-s + 67-s + 2·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5517801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5517801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8999858439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8999858439\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 29 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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
−9.446295129169520255240476262996, −8.910984258929684722708219687622, −8.412138299847160747349755640002, −8.320865341611171081410594024599, −7.79390817150058365486026692100, −7.74838802162324416450205172911, −7.28923919052202620910187721647, −6.51022253449871771757361718452, −6.45021212996510603166879184992, −5.71507915810527708854598076809, −5.63913720750977585500279247924, −5.26206935524964163026334869826, −4.67572823611261978021256647948, −3.86887968909074910615004539819, −3.68815216195576276136331921070, −2.91310204240849799603633423801, −2.73437569766984234489949647826, −2.06882255985564483004333347178, −1.37307258523795463114404955882, −0.837912564029636181169903870958,
0.837912564029636181169903870958, 1.37307258523795463114404955882, 2.06882255985564483004333347178, 2.73437569766984234489949647826, 2.91310204240849799603633423801, 3.68815216195576276136331921070, 3.86887968909074910615004539819, 4.67572823611261978021256647948, 5.26206935524964163026334869826, 5.63913720750977585500279247924, 5.71507915810527708854598076809, 6.45021212996510603166879184992, 6.51022253449871771757361718452, 7.28923919052202620910187721647, 7.74838802162324416450205172911, 7.79390817150058365486026692100, 8.320865341611171081410594024599, 8.412138299847160747349755640002, 8.910984258929684722708219687622, 9.446295129169520255240476262996
