| L(s) = 1 | + 16·2-s − 119·3-s + 256·4-s − 684·5-s − 1.90e3·6-s + 9.14e3·7-s + 4.09e3·8-s − 5.52e3·9-s − 1.09e4·10-s + 5.79e3·11-s − 3.04e4·12-s − 1.79e5·13-s + 1.46e5·14-s + 8.13e4·15-s + 6.55e4·16-s − 5.94e5·17-s − 8.83e4·18-s + 1.30e5·19-s − 1.75e5·20-s − 1.08e6·21-s + 9.26e4·22-s − 1.74e6·23-s − 4.87e5·24-s − 1.48e6·25-s − 2.87e6·26-s + 2.99e6·27-s + 2.34e6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.848·3-s + 1/2·4-s − 0.489·5-s − 0.599·6-s + 1.44·7-s + 0.353·8-s − 0.280·9-s − 0.346·10-s + 0.119·11-s − 0.424·12-s − 1.74·13-s + 1.01·14-s + 0.415·15-s + 1/4·16-s − 1.72·17-s − 0.198·18-s + 0.229·19-s − 0.244·20-s − 1.22·21-s + 0.0843·22-s − 1.30·23-s − 0.299·24-s − 0.760·25-s − 1.23·26-s + 1.08·27-s + 0.720·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{4} T \) |
| 19 | \( 1 - p^{4} T \) |
| good | 3 | \( 1 + 119 T + p^{9} T^{2} \) |
| 5 | \( 1 + 684 T + p^{9} T^{2} \) |
| 7 | \( 1 - 1307 p T + p^{9} T^{2} \) |
| 11 | \( 1 - 5790 T + p^{9} T^{2} \) |
| 13 | \( 1 + 13837 p T + p^{9} T^{2} \) |
| 17 | \( 1 + 594093 T + p^{9} T^{2} \) |
| 23 | \( 1 + 1744767 T + p^{9} T^{2} \) |
| 29 | \( 1 - 4314387 T + p^{9} T^{2} \) |
| 31 | \( 1 - 160232 T + p^{9} T^{2} \) |
| 37 | \( 1 + 21943090 T + p^{9} T^{2} \) |
| 41 | \( 1 - 294816 T + p^{9} T^{2} \) |
| 43 | \( 1 + 39393148 T + p^{9} T^{2} \) |
| 47 | \( 1 - 46596360 T + p^{9} T^{2} \) |
| 53 | \( 1 - 22121703 T + p^{9} T^{2} \) |
| 59 | \( 1 - 33070233 T + p^{9} T^{2} \) |
| 61 | \( 1 - 188535938 T + p^{9} T^{2} \) |
| 67 | \( 1 + 20769067 T + p^{9} T^{2} \) |
| 71 | \( 1 + 232299978 T + p^{9} T^{2} \) |
| 73 | \( 1 + 3022183 T + p^{9} T^{2} \) |
| 79 | \( 1 + 446379406 T + p^{9} T^{2} \) |
| 83 | \( 1 - 794022846 T + p^{9} T^{2} \) |
| 89 | \( 1 - 90999336 T + p^{9} T^{2} \) |
| 97 | \( 1 + 123974170 T + p^{9} T^{2} \) |
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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
−13.84450282785956758063480273329, −12.00238953690549248423450897486, −11.70913940370908431163549907404, −10.45924874595720139270497113509, −8.320245388567686455275645635525, −6.93660421583044376162866144726, −5.30120767101136794938992542141, −4.41562816141538458063004021800, −2.15173522092135183904120949750, 0,
2.15173522092135183904120949750, 4.41562816141538458063004021800, 5.30120767101136794938992542141, 6.93660421583044376162866144726, 8.320245388567686455275645635525, 10.45924874595720139270497113509, 11.70913940370908431163549907404, 12.00238953690549248423450897486, 13.84450282785956758063480273329
