| L(s) = 1 | + 5-s + 9-s − 19-s + 29-s + 2·41-s + 45-s − 2·49-s + 3·59-s − 61-s − 3·71-s + 3·79-s + 89-s − 95-s + 101-s − 109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 5-s + 9-s − 19-s + 29-s + 2·41-s + 45-s − 2·49-s + 3·59-s − 61-s − 3·71-s + 3·79-s + 89-s − 95-s + 101-s − 109-s − 121-s − 125-s + 127-s + 131-s + 137-s + 139-s + 145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.490338702\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.490338702\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 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
−9.893156799054086941989927713626, −9.368432600161537641113979317941, −9.290195944824512957693638488847, −8.706008666461950198559937545803, −8.226735079712346508192466413331, −7.969995004651745248940009322139, −7.31351995659856368094233694782, −7.12888607126374888029364825173, −6.48627451305059621222529688275, −6.20980231256534698137159676296, −5.95129878512687378939338463437, −5.32098979218182876176465308713, −4.74209499115937212588253964221, −4.59352821367347328190114606253, −3.89503670198133601378328688253, −3.55235791243882900081794795775, −2.64063483348644297480854123480, −2.36949069645953042159712633249, −1.69876312402771364312173726435, −1.09378534532884611647703474898,
1.09378534532884611647703474898, 1.69876312402771364312173726435, 2.36949069645953042159712633249, 2.64063483348644297480854123480, 3.55235791243882900081794795775, 3.89503670198133601378328688253, 4.59352821367347328190114606253, 4.74209499115937212588253964221, 5.32098979218182876176465308713, 5.95129878512687378939338463437, 6.20980231256534698137159676296, 6.48627451305059621222529688275, 7.12888607126374888029364825173, 7.31351995659856368094233694782, 7.969995004651745248940009322139, 8.226735079712346508192466413331, 8.706008666461950198559937545803, 9.290195944824512957693638488847, 9.368432600161537641113979317941, 9.893156799054086941989927713626
