| L(s) = 1 | − 4·2-s + 3-s + 12·4-s + 10·5-s − 4·6-s + 57·7-s − 32·8-s − 9·9-s − 40·10-s + 10·11-s + 12·12-s + 13·13-s − 228·14-s + 10·15-s + 80·16-s − 51·17-s + 36·18-s − 38·19-s + 120·20-s + 57·21-s − 40·22-s − 155·23-s − 32·24-s + 2·25-s − 52·26-s + 8·27-s + 684·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.192·3-s + 3/2·4-s + 0.894·5-s − 0.272·6-s + 3.07·7-s − 1.41·8-s − 1/3·9-s − 1.26·10-s + 0.274·11-s + 0.288·12-s + 0.277·13-s − 4.35·14-s + 0.172·15-s + 5/4·16-s − 0.727·17-s + 0.471·18-s − 0.458·19-s + 1.34·20-s + 0.592·21-s − 0.387·22-s − 1.40·23-s − 0.272·24-s + 0.0159·25-s − 0.392·26-s + 0.0570·27-s + 4.61·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.261919795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261919795\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 10 T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 2 p T + 98 T^{2} - 2 p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 57 T + 1454 T^{2} - 57 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 10 T + 2510 T^{2} - 10 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - p T + 2268 T^{2} - p^{4} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 p T + 520 T^{2} + 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 155 T + 994 p T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 79 T + 13124 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 16 T + 48318 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 380 T + 126078 T^{2} - 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 790 T + 292274 T^{2} + 790 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 296 T + 78966 T^{2} - 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 200 T + 146846 T^{2} + 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 397 T + 333572 T^{2} - 397 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 201 T + 197794 T^{2} - 201 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 680 T + 483894 T^{2} + 680 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 939 T + 740138 T^{2} + 939 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 406 T + 735614 T^{2} - 406 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 123 T + 781772 T^{2} - 123 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 106 T + 840030 T^{2} - 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2226 T + 2380750 T^{2} - 2226 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 870 T + 1594738 T^{2} + 870 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1864 T + 2438382 T^{2} + 1864 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−16.42362552439556254127211450200, −15.44249293269138385164643740783, −14.83341150220674979248969608990, −14.65462673527854666915208766640, −13.84081462250573773073310298309, −13.42805270694816707438894980498, −11.93686823526188039545269815241, −11.81518792602670282986256710435, −10.89036109132546340838310197267, −10.80785125592809109236580824559, −9.841794763631763200053699624608, −9.072963783034075895055675876009, −8.266695078387097207180441493137, −8.232195932100640022413740604025, −7.35742288151623467179702724045, −6.24615267287495912324842548824, −5.39204263415166716592434760149, −4.37425361656522483900048009165, −2.14164103742936405307762083682, −1.58422284959852460289428135042,
1.58422284959852460289428135042, 2.14164103742936405307762083682, 4.37425361656522483900048009165, 5.39204263415166716592434760149, 6.24615267287495912324842548824, 7.35742288151623467179702724045, 8.232195932100640022413740604025, 8.266695078387097207180441493137, 9.072963783034075895055675876009, 9.841794763631763200053699624608, 10.80785125592809109236580824559, 10.89036109132546340838310197267, 11.81518792602670282986256710435, 11.93686823526188039545269815241, 13.42805270694816707438894980498, 13.84081462250573773073310298309, 14.65462673527854666915208766640, 14.83341150220674979248969608990, 15.44249293269138385164643740783, 16.42362552439556254127211450200
