| L(s) = 1 | − 6·3-s − 6·5-s − 14·7-s + 27·9-s + 26·11-s + 96·13-s + 36·15-s + 78·17-s + 40·19-s + 84·21-s + 22·23-s + 114·25-s − 108·27-s + 204·29-s + 96·31-s − 156·33-s + 84·35-s + 504·37-s − 576·39-s − 102·41-s + 296·43-s − 162·45-s − 780·47-s + 147·49-s − 468·51-s + 192·53-s − 156·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.536·5-s − 0.755·7-s + 9-s + 0.712·11-s + 2.04·13-s + 0.619·15-s + 1.11·17-s + 0.482·19-s + 0.872·21-s + 0.199·23-s + 0.911·25-s − 0.769·27-s + 1.30·29-s + 0.556·31-s − 0.822·33-s + 0.405·35-s + 2.23·37-s − 2.36·39-s − 0.388·41-s + 1.04·43-s − 0.536·45-s − 2.42·47-s + 3/7·49-s − 1.28·51-s + 0.497·53-s − 0.382·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.672359083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672359083\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 6 T - 78 T^{2} + 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 2494 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 96 T + 5350 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 78 T + 8314 T^{2} - 78 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $D_{4}$ | \( 1 - 22 T + 16030 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 102 T + p^{3} T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 96 T - 24386 T^{2} - 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 504 T + 163462 T^{2} - 504 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 102 T + 99666 T^{2} + 102 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 296 T + 94646 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 780 T + 326046 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 192 T + 197782 T^{2} - 192 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 212 T + 355942 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 100 T + 370190 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 212 T + 611414 T^{2} - 212 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 534 T + 746334 T^{2} + 534 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1128 T + 1062430 T^{2} - 1128 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 468 T - 92834 T^{2} - 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 824 T + 536870 T^{2} + 824 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 2118 T + 2285746 T^{2} + 2118 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 400 T + 214046 T^{2} - 400 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.41449189636792227017801301889, −12.12251618646484611282803624573, −11.36856582385857406340867699665, −11.31419035895456360031854173681, −10.76557959130858921961004318317, −10.04688358149399625939756750083, −9.735415811953106509282888575549, −9.122028349312174127067240093275, −8.251077222672277044720048558403, −8.152289422923133393697136870565, −7.07890859885210978147211196784, −6.73720732022139089823631173399, −6.01779464271020646140084335224, −5.91081866313314362368666823592, −4.91091511713861246179189234007, −4.25897752644496606387226657173, −3.58276926273752872821345606716, −2.96574736656808593985663017218, −1.21895407688148780431106327794, −0.806139301870196116638499140698,
0.806139301870196116638499140698, 1.21895407688148780431106327794, 2.96574736656808593985663017218, 3.58276926273752872821345606716, 4.25897752644496606387226657173, 4.91091511713861246179189234007, 5.91081866313314362368666823592, 6.01779464271020646140084335224, 6.73720732022139089823631173399, 7.07890859885210978147211196784, 8.152289422923133393697136870565, 8.251077222672277044720048558403, 9.122028349312174127067240093275, 9.735415811953106509282888575549, 10.04688358149399625939756750083, 10.76557959130858921961004318317, 11.31419035895456360031854173681, 11.36856582385857406340867699665, 12.12251618646484611282803624573, 12.41449189636792227017801301889
