| L(s) = 1 | − 32·2-s + 162·3-s + 768·4-s + 104·5-s − 5.18e3·6-s + 6.10e3·7-s − 1.63e4·8-s + 1.96e4·9-s − 3.32e3·10-s + 3.85e4·11-s + 1.24e5·12-s − 5.71e4·13-s − 1.95e5·14-s + 1.68e4·15-s + 3.27e5·16-s + 6.34e5·17-s − 6.29e5·18-s + 7.60e5·19-s + 7.98e4·20-s + 9.88e5·21-s − 1.23e6·22-s − 1.47e4·23-s − 2.65e6·24-s − 3.85e6·25-s + 1.82e6·26-s + 2.12e6·27-s + 4.68e6·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.0744·5-s − 1.63·6-s + 0.960·7-s − 1.41·8-s + 9-s − 0.105·10-s + 0.793·11-s + 1.73·12-s − 0.554·13-s − 1.35·14-s + 0.0859·15-s + 5/4·16-s + 1.84·17-s − 1.41·18-s + 1.33·19-s + 0.111·20-s + 1.10·21-s − 1.12·22-s − 0.0109·23-s − 1.63·24-s − 1.97·25-s + 0.784·26-s + 0.769·27-s + 1.44·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6084 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.839251362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839251362\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + p^{4} T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 - 104 T + 773626 p T^{2} - 104 p^{9} T^{3} + p^{18} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 872 p T + 199686 p^{2} T^{2} - 872 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 38524 T + 3497215322 T^{2} - 38524 p^{9} T^{3} + p^{18} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 634548 T + 312873688294 T^{2} - 634548 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 760680 T + 488815383422 T^{2} - 760680 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 14728 T + 3591989602238 T^{2} + 14728 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 41596 T + 27363410038142 T^{2} + 41596 p^{9} T^{3} + p^{18} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 7634224 T + 2155903210170 p T^{2} + 7634224 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4857588 T - 109643711805106 T^{2} + 4857588 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12963768 T + 335840336364778 T^{2} - 12963768 p^{9} T^{3} + p^{18} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 26477144 T + 471173894566470 T^{2} - 26477144 p^{9} T^{3} + p^{18} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 1195196 p T + 2892881658398594 T^{2} - 1195196 p^{10} T^{3} + p^{18} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 13129596 T + 1481999213333806 T^{2} - 13129596 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 304382900 T + 40398801995110778 T^{2} - 304382900 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 131387684 T + 27551754684980862 T^{2} - 131387684 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 271132832 T + 69453288411977550 T^{2} - 271132832 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 495302596 T + 151357851213379730 T^{2} - 495302596 p^{9} T^{3} + p^{18} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 34347812 T + 31939863679238262 T^{2} + 34347812 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 602655392 T + 330078647490497310 T^{2} + 602655392 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 303013932 T + 241347296159168362 T^{2} - 303013932 p^{9} T^{3} + p^{18} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 703580304 T + 815686477875577978 T^{2} + 703580304 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 189239212 T + 1528326062462815686 T^{2} - 189239212 p^{9} T^{3} + p^{18} T^{4} \) |
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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
−12.57739792930751394962650377344, −12.27056273529313650404091644830, −11.41900153967180827722619140223, −11.37884486656022209535867431354, −10.24227151673835109807332567313, −9.931364959933567183254863482590, −9.419110713128219070991093949120, −9.037502967093529007531268731115, −8.140057873370876983961620575286, −8.003369336116575554255280768647, −7.20553917181006016811103198082, −7.12966286447917962446436156289, −5.63047159840732971704941817036, −5.45425573985342767716994171652, −3.90676773008896713733085349664, −3.62888390804358743204669220236, −2.45977676685239845088034113540, −2.03638720310041338501972443373, −1.17272655831602423209132361591, −0.75898284019281359581146040316,
0.75898284019281359581146040316, 1.17272655831602423209132361591, 2.03638720310041338501972443373, 2.45977676685239845088034113540, 3.62888390804358743204669220236, 3.90676773008896713733085349664, 5.45425573985342767716994171652, 5.63047159840732971704941817036, 7.12966286447917962446436156289, 7.20553917181006016811103198082, 8.003369336116575554255280768647, 8.140057873370876983961620575286, 9.037502967093529007531268731115, 9.419110713128219070991093949120, 9.931364959933567183254863482590, 10.24227151673835109807332567313, 11.37884486656022209535867431354, 11.41900153967180827722619140223, 12.27056273529313650404091644830, 12.57739792930751394962650377344
