| L(s) = 1 | + 2·3-s + 6·7-s − 9-s − 6·13-s − 2·17-s + 2·19-s + 12·21-s + 10·23-s − 6·27-s − 6·29-s − 4·31-s − 12·39-s + 12·43-s + 15·49-s − 4·51-s + 6·53-s + 4·57-s + 6·59-s + 20·61-s − 6·63-s + 18·67-s + 20·69-s + 24·71-s − 6·73-s − 4·79-s − 4·81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s − 1/3·9-s − 1.66·13-s − 0.485·17-s + 0.458·19-s + 2.61·21-s + 2.08·23-s − 1.15·27-s − 1.11·29-s − 0.718·31-s − 1.92·39-s + 1.82·43-s + 15/7·49-s − 0.560·51-s + 0.824·53-s + 0.529·57-s + 0.781·59-s + 2.56·61-s − 0.755·63-s + 2.19·67-s + 2.40·69-s + 2.84·71-s − 0.702·73-s − 0.450·79-s − 4/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57760000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.975143310\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.975143310\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 6 T + 3 p T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6 T + 27 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 53 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 104 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 29 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 20 T + 204 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 18 T + 197 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 24 T + 4 p T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6 T + 83 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_4$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.976584807959091217271511858297, −7.82016010939966412617273180562, −7.36243594287606316923766247107, −7.28557645294643647943438977413, −6.85083730224638853489088562510, −6.48639967789367723116518139786, −5.61399051349855240940904118844, −5.57449215302970781917805708784, −5.08528312481028799144300779478, −5.00233746173306346607435648553, −4.60830474472460529232582775294, −4.10994176468701710379574596263, −3.56843083734713254058273416968, −3.46783207734431914887480932850, −2.62057138152629032964333986600, −2.48829895444736018691039011798, −2.06754149125574246855682069547, −1.89270961611857388709543818137, −0.932759353704666590548488436690, −0.67747242566603682892335784678,
0.67747242566603682892335784678, 0.932759353704666590548488436690, 1.89270961611857388709543818137, 2.06754149125574246855682069547, 2.48829895444736018691039011798, 2.62057138152629032964333986600, 3.46783207734431914887480932850, 3.56843083734713254058273416968, 4.10994176468701710379574596263, 4.60830474472460529232582775294, 5.00233746173306346607435648553, 5.08528312481028799144300779478, 5.57449215302970781917805708784, 5.61399051349855240940904118844, 6.48639967789367723116518139786, 6.85083730224638853489088562510, 7.28557645294643647943438977413, 7.36243594287606316923766247107, 7.82016010939966412617273180562, 7.976584807959091217271511858297
