L(s) = 1 | − 2-s − 3-s − 4-s − 4·5-s + 6-s − 2·7-s + 3·8-s − 2·9-s + 4·10-s − 6·11-s + 12-s − 4·13-s + 2·14-s + 4·15-s − 16-s + 4·17-s + 2·18-s − 6·19-s + 4·20-s + 2·21-s + 6·22-s − 9·23-s − 3·24-s + 11·25-s + 4·26-s + 5·27-s + 2·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 1.78·5-s + 0.408·6-s − 0.755·7-s + 1.06·8-s − 2/3·9-s + 1.26·10-s − 1.80·11-s + 0.288·12-s − 1.10·13-s + 0.534·14-s + 1.03·15-s − 1/4·16-s + 0.970·17-s + 0.471·18-s − 1.37·19-s + 0.894·20-s + 0.436·21-s + 1.27·22-s − 1.87·23-s − 0.612·24-s + 11/5·25-s + 0.784·26-s + 0.962·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21443 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21443 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 41 | \( 1 + T \) |
| 523 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−16.27158432621510, −16.16812998016164, −15.42058473329263, −14.86570491258612, −14.37449062146839, −13.61587628049915, −12.84406631445973, −12.38153333104854, −12.25004310186594, −11.31637945346990, −10.62972844130414, −10.60702657183148, −9.772390097068296, −9.149380734138057, −8.396584224189151, −8.010118780443887, −7.477951619368623, −7.215701678589239, −6.022131891199877, −5.470639130593806, −4.846815847728758, −4.176822936923978, −3.532549934632547, −2.843531800945688, −1.818837492966736, 0, 0, 0,
1.818837492966736, 2.843531800945688, 3.532549934632547, 4.176822936923978, 4.846815847728758, 5.470639130593806, 6.022131891199877, 7.215701678589239, 7.477951619368623, 8.010118780443887, 8.396584224189151, 9.149380734138057, 9.772390097068296, 10.60702657183148, 10.62972844130414, 11.31637945346990, 12.25004310186594, 12.38153333104854, 12.84406631445973, 13.61587628049915, 14.37449062146839, 14.86570491258612, 15.42058473329263, 16.16812998016164, 16.27158432621510