L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 3·7-s + 8-s + 9-s − 10-s + 4·11-s − 12-s − 3·13-s + 3·14-s + 15-s + 16-s + 18-s − 7·19-s − 20-s − 3·21-s + 4·22-s − 4·23-s − 24-s + 25-s − 3·26-s − 27-s + 3·28-s − 29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.832·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 1.60·19-s − 0.223·20-s − 0.654·21-s + 0.852·22-s − 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s − 0.192·27-s + 0.566·28-s − 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 43 | \( 1 \) |
good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.59089169736837, −14.21745527523905, −13.82876074024168, −13.00951551962530, −12.45068197510350, −11.99046781178824, −11.89395852531358, −11.08405725782484, −10.78893248818596, −10.30932056459259, −9.495289490149173, −8.941101171716305, −8.273895885263147, −7.809770717634672, −7.197870517859948, −6.649735120302737, −6.157896834642827, −5.563092628645392, −4.774911792674070, −4.532822570120022, −3.980721991187328, −3.373117510723612, −2.259513461561813, −1.890097436088680, −1.046363977428153, 0,
1.046363977428153, 1.890097436088680, 2.259513461561813, 3.373117510723612, 3.980721991187328, 4.532822570120022, 4.774911792674070, 5.563092628645392, 6.157896834642827, 6.649735120302737, 7.197870517859948, 7.809770717634672, 8.273895885263147, 8.941101171716305, 9.495289490149173, 10.30932056459259, 10.78893248818596, 11.08405725782484, 11.89395852531358, 11.99046781178824, 12.45068197510350, 13.00951551962530, 13.82876074024168, 14.21745527523905, 14.59089169736837