| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 11-s − 13-s − 16-s − 2·17-s + 22-s + 23-s + 26-s − 29-s + 31-s + 2·34-s − 4·37-s + 40-s − 43-s − 46-s + 47-s − 49-s − 55-s + 58-s + 2·59-s − 62-s + 64-s − 65-s + 2·67-s + ⋯ |
| L(s) = 1 | − 2-s + 5-s + 8-s − 10-s − 11-s − 13-s − 16-s − 2·17-s + 22-s + 23-s + 26-s − 29-s + 31-s + 2·34-s − 4·37-s + 40-s − 43-s − 46-s + 47-s − 49-s − 55-s + 58-s + 2·59-s − 62-s + 64-s − 65-s + 2·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4296834874\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4296834874\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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_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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 37 | $C_1$ | \( ( 1 + T )^{4} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 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_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−9.067042929703856636288672322902, −8.536995632400850000115093830754, −8.453653159677570187174207579985, −8.186921106169253930510932042461, −7.38973233283220143214869712989, −7.11382867356695623465213506854, −7.09740091834331479366786297305, −6.42039318673403441028219751063, −6.25646073826014993535611414877, −5.29391189981654398968661497272, −5.16384300675554929154509548313, −5.10179098569669817114853987564, −4.56780495873831782646466257111, −3.74451570577425587629406938305, −3.69247992007079660168033112125, −2.61451903850541710074882236799, −2.50195402673133539025340630028, −1.85718105529215349721674279729, −1.62701644104950975271333890849, −0.44841016151069281578945038242,
0.44841016151069281578945038242, 1.62701644104950975271333890849, 1.85718105529215349721674279729, 2.50195402673133539025340630028, 2.61451903850541710074882236799, 3.69247992007079660168033112125, 3.74451570577425587629406938305, 4.56780495873831782646466257111, 5.10179098569669817114853987564, 5.16384300675554929154509548313, 5.29391189981654398968661497272, 6.25646073826014993535611414877, 6.42039318673403441028219751063, 7.09740091834331479366786297305, 7.11382867356695623465213506854, 7.38973233283220143214869712989, 8.186921106169253930510932042461, 8.453653159677570187174207579985, 8.536995632400850000115093830754, 9.067042929703856636288672322902
