| L(s) = 1 | + 8·3-s + 10·5-s + 8·7-s + 18·9-s + 64·11-s + 12·13-s + 80·15-s + 4·17-s + 208·19-s + 64·21-s + 120·23-s + 75·25-s − 8·27-s − 292·29-s + 176·31-s + 512·33-s + 80·35-s − 356·37-s + 96·39-s + 100·41-s − 376·43-s + 180·45-s − 280·47-s − 38·49-s + 32·51-s + 316·53-s + 640·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.894·5-s + 0.431·7-s + 2/3·9-s + 1.75·11-s + 0.256·13-s + 1.37·15-s + 0.0570·17-s + 2.51·19-s + 0.665·21-s + 1.08·23-s + 3/5·25-s − 0.0570·27-s − 1.86·29-s + 1.01·31-s + 2.70·33-s + 0.386·35-s − 1.58·37-s + 0.394·39-s + 0.380·41-s − 1.33·43-s + 0.596·45-s − 0.868·47-s − 0.110·49-s + 0.0878·51-s + 0.818·53-s + 1.56·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25600 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.925769076\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.925769076\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 64 T + 2822 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 12 T + 2894 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T - 3994 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 208 T + 22998 T^{2} - 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 120 T + 25030 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 292 T + 63950 T^{2} + 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 176 T + 66462 T^{2} - 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 356 T + 126846 T^{2} + 356 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 100 T + 101942 T^{2} - 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 376 T + 161502 T^{2} + 376 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 280 T + 223190 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 316 T + 308894 T^{2} - 316 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 720 T + 530758 T^{2} - 720 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1268 T + 831342 T^{2} + 1268 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 744 T + 686894 T^{2} + 744 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 48 T - 424178 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 940 T + 653334 T^{2} + 940 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 32 T + 276318 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1592 T + 1724174 T^{2} + 1592 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 780 T + 1506742 T^{2} + 780 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1220 T + 2159046 T^{2} - 1220 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.90057976641195194326403702788, −12.07944666155499279161735114316, −11.59454307411752076025443783844, −11.43786892411146527381082739595, −10.48909510800178765538128060999, −9.946914282481845073682175216502, −9.374549824180792744869599810321, −9.132337644566011188498566344507, −8.724896961806542289458338668585, −8.200018756500394738665247039334, −7.32039364854479990015315683827, −7.14390397592921254683681862828, −6.28480277508495431318534754157, −5.57625104557087285003188415581, −5.01782670845119088910440796014, −4.06655852415350657571465781181, −3.19481007479969433618600058947, −3.02617259156789579275513362149, −1.70856544325521916667768543424, −1.29698493764042189011672068702,
1.29698493764042189011672068702, 1.70856544325521916667768543424, 3.02617259156789579275513362149, 3.19481007479969433618600058947, 4.06655852415350657571465781181, 5.01782670845119088910440796014, 5.57625104557087285003188415581, 6.28480277508495431318534754157, 7.14390397592921254683681862828, 7.32039364854479990015315683827, 8.200018756500394738665247039334, 8.724896961806542289458338668585, 9.132337644566011188498566344507, 9.374549824180792744869599810321, 9.946914282481845073682175216502, 10.48909510800178765538128060999, 11.43786892411146527381082739595, 11.59454307411752076025443783844, 12.07944666155499279161735114316, 12.90057976641195194326403702788
