| L(s) = 1 | + 2-s + 3-s + 4-s − 4·5-s + 6-s + 8-s + 9-s − 4·10-s − 2·11-s + 12-s + 13-s − 4·15-s + 16-s − 3·17-s + 18-s + 19-s − 4·20-s − 2·22-s + 8·23-s + 24-s + 11·25-s + 26-s + 27-s − 5·29-s − 4·30-s + 32-s − 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.603·11-s + 0.288·12-s + 0.277·13-s − 1.03·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s − 0.894·20-s − 0.426·22-s + 1.66·23-s + 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s − 0.928·29-s − 0.730·30-s + 0.176·32-s − 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.12682721423330, −15.87078325939549, −15.41726971000816, −14.80168943173835, −14.60328466100106, −13.62334480018417, −13.09914008718659, −12.75558647681386, −11.97220576206417, −11.56556573496647, −10.78456427748478, −10.70394147430753, −9.501534186621947, −8.814021970510102, −8.322394145859345, −7.676342423870858, −7.123424353217182, −6.763765328551489, −5.596192249765694, −4.917946711960394, −4.309158382767921, −3.671543562682198, −3.143902714792727, −2.431162537021674, −1.208356875968021, 0,
1.208356875968021, 2.431162537021674, 3.143902714792727, 3.671543562682198, 4.309158382767921, 4.917946711960394, 5.596192249765694, 6.763765328551489, 7.123424353217182, 7.676342423870858, 8.322394145859345, 8.814021970510102, 9.501534186621947, 10.70394147430753, 10.78456427748478, 11.56556573496647, 11.97220576206417, 12.75558647681386, 13.09914008718659, 13.62334480018417, 14.60328466100106, 14.80168943173835, 15.41726971000816, 15.87078325939549, 16.12682721423330
