| L(s) = 1 | − 3-s − 9-s − 2·11-s − 5·13-s − 2·17-s − 2·19-s − 2·23-s + 3·29-s + 9·31-s + 2·33-s + 5·39-s + 41-s − 16·43-s + 11·47-s − 14·49-s + 2·51-s − 4·53-s + 2·57-s − 4·59-s + 8·61-s − 2·67-s + 2·69-s − 23·71-s + 17·73-s + 2·79-s − 4·81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/3·9-s − 0.603·11-s − 1.38·13-s − 0.485·17-s − 0.458·19-s − 0.417·23-s + 0.557·29-s + 1.61·31-s + 0.348·33-s + 0.800·39-s + 0.156·41-s − 2.43·43-s + 1.60·47-s − 2·49-s + 0.280·51-s − 0.549·53-s + 0.264·57-s − 0.520·59-s + 1.02·61-s − 0.244·67-s + 0.240·69-s − 2.72·71-s + 1.98·73-s + 0.225·79-s − 4/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 + 2 T + 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T - 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 11 T + 86 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 23 T + 270 T^{2} + 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.45169725734863801081418939902, −7.38085088881450108956215160780, −6.64765505834166374718954047462, −6.54091552914224000872917368841, −6.26407507417887905991978890288, −5.91852323016016111224414218565, −5.35726456235825819391479947269, −5.14649185466780686155161355455, −4.76244057870749514570948533916, −4.58772545784439561383147201391, −4.19565229233611241609677023553, −3.57341092818981216361706786433, −3.19091338188650723765823553828, −2.84983340448071890250094770966, −2.25389691794772682973780894451, −2.19019648442119675153567815300, −1.47916526849993132058537150080, −0.852952180139238681139519921718, 0, 0,
0.852952180139238681139519921718, 1.47916526849993132058537150080, 2.19019648442119675153567815300, 2.25389691794772682973780894451, 2.84983340448071890250094770966, 3.19091338188650723765823553828, 3.57341092818981216361706786433, 4.19565229233611241609677023553, 4.58772545784439561383147201391, 4.76244057870749514570948533916, 5.14649185466780686155161355455, 5.35726456235825819391479947269, 5.91852323016016111224414218565, 6.26407507417887905991978890288, 6.54091552914224000872917368841, 6.64765505834166374718954047462, 7.38085088881450108956215160780, 7.45169725734863801081418939902
