| L(s) = 1 | − 16·2-s − 54·3-s + 192·4-s + 250·5-s + 864·6-s − 686·7-s − 2.04e3·8-s + 2.18e3·9-s − 4.00e3·10-s + 2.09e3·11-s − 1.03e4·12-s + 1.09e3·13-s + 1.09e4·14-s − 1.35e4·15-s + 2.04e4·16-s + 2.26e3·17-s − 3.49e4·18-s − 6.48e4·19-s + 4.80e4·20-s + 3.70e4·21-s − 3.35e4·22-s + 3.97e4·23-s + 1.10e5·24-s + 4.68e4·25-s − 1.74e4·26-s − 7.87e4·27-s − 1.31e5·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 0.755·7-s − 1.41·8-s + 9-s − 1.26·10-s + 0.474·11-s − 1.73·12-s + 0.137·13-s + 1.06·14-s − 1.03·15-s + 5/4·16-s + 0.111·17-s − 1.41·18-s − 2.17·19-s + 1.34·20-s + 0.872·21-s − 0.671·22-s + 0.681·23-s + 1.63·24-s + 3/5·25-s − 0.194·26-s − 0.769·27-s − 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 2096 T + 24040262 T^{2} - 2096 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 84 p T + 25592750 T^{2} - 84 p^{8} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2260 T + 785881382 T^{2} - 2260 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 64896 T + 2584098038 T^{2} + 64896 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1728 p T - 206422226 T^{2} - 1728 p^{8} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 120084 T + 7767515758 T^{2} + 120084 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 103112 T + 52196165934 T^{2} + 103112 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 856780 T + 371934823710 T^{2} - 856780 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 603844 T + 217990862390 T^{2} - 603844 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 784760 T + 658315937718 T^{2} - 784760 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 522448 T + 1077119402558 T^{2} - 522448 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 683828 T + 247702035614 T^{2} - 683828 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 83512 T + 1718043595190 T^{2} + 83512 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2857444 T + 6254702909070 T^{2} + 2857444 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 841656 T + 12164292289286 T^{2} - 841656 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 7188152 T + 30019661024222 T^{2} + 7188152 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 466540 T - 1772103012810 T^{2} - 466540 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5886256 T + 34597844105886 T^{2} + 5886256 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 11940392 T + 88364674044806 T^{2} + 11940392 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 11284060 T + 111228795030422 T^{2} + 11284060 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15217556 T + 129095478513414 T^{2} + 15217556 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.61669872553177412913250279369, −10.55640304278340823157910230080, −9.713334526307873298380211253896, −9.549194262194495021887002071304, −8.938996945666565946211644042871, −8.694477691362786834419423499403, −7.64883315452395699080773528573, −7.40714386686086670674505739159, −6.62083664213157938318168351422, −6.35557496861686060686018742399, −5.78019632600603608157201676950, −5.61628356317094006628183610095, −4.21522595196974958923741900126, −4.16829949728990708990717687571, −2.59099240551909084550170080178, −2.56632868747299567737646935580, −1.26396800511561068462896280714, −1.24995492586745947970338332680, 0, 0,
1.24995492586745947970338332680, 1.26396800511561068462896280714, 2.56632868747299567737646935580, 2.59099240551909084550170080178, 4.16829949728990708990717687571, 4.21522595196974958923741900126, 5.61628356317094006628183610095, 5.78019632600603608157201676950, 6.35557496861686060686018742399, 6.62083664213157938318168351422, 7.40714386686086670674505739159, 7.64883315452395699080773528573, 8.694477691362786834419423499403, 8.938996945666565946211644042871, 9.549194262194495021887002071304, 9.713334526307873298380211253896, 10.55640304278340823157910230080, 10.61669872553177412913250279369
