| L(s) = 1 | − 3·2-s + 4·4-s − 2·7-s − 3·8-s + 2·11-s + 6·13-s + 6·14-s + 3·16-s + 2·17-s − 2·19-s − 6·22-s − 10·23-s − 18·26-s − 8·28-s + 4·29-s − 6·32-s − 6·34-s + 2·37-s + 6·38-s + 16·41-s + 12·43-s + 8·44-s + 30·46-s − 10·47-s + 3·49-s + 24·52-s + 8·53-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 0.755·7-s − 1.06·8-s + 0.603·11-s + 1.66·13-s + 1.60·14-s + 3/4·16-s + 0.485·17-s − 0.458·19-s − 1.27·22-s − 2.08·23-s − 3.53·26-s − 1.51·28-s + 0.742·29-s − 1.06·32-s − 1.02·34-s + 0.328·37-s + 0.973·38-s + 2.49·41-s + 1.82·43-s + 1.20·44-s + 4.42·46-s − 1.45·47-s + 3/7·49-s + 3.32·52-s + 1.09·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7220477129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7220477129\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T - 10 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 34 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 17 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 55 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 117 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 122 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_4$ | \( 1 + 4 T - 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 105 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 10 T + 147 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 169 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 210 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2 T + 134 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 16 T + 238 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−9.445926743058312038995117615854, −9.348173658778707124473277094149, −8.792025544568502473354343280102, −8.548379624885392315692319611720, −8.172581544625168907373866503354, −7.898763926742529603107324079522, −7.38422640039892577952146538504, −7.07648225346657981656306154729, −6.26172044559035611107708044191, −6.11587872576812019024715464593, −6.01101416446183916950254835053, −5.31400977333011272118104632219, −4.32591314186731984488154729504, −4.13465388679144480844576426172, −3.57727275723253702872333039579, −3.09296003911675803365689704610, −2.27657767780759479104994001238, −1.80571237854845747522753979069, −0.851799364962200221279542813047, −0.70466716119547569152274873109,
0.70466716119547569152274873109, 0.851799364962200221279542813047, 1.80571237854845747522753979069, 2.27657767780759479104994001238, 3.09296003911675803365689704610, 3.57727275723253702872333039579, 4.13465388679144480844576426172, 4.32591314186731984488154729504, 5.31400977333011272118104632219, 6.01101416446183916950254835053, 6.11587872576812019024715464593, 6.26172044559035611107708044191, 7.07648225346657981656306154729, 7.38422640039892577952146538504, 7.898763926742529603107324079522, 8.172581544625168907373866503354, 8.548379624885392315692319611720, 8.792025544568502473354343280102, 9.348173658778707124473277094149, 9.445926743058312038995117615854
