| L(s) = 1 | + 4.09e3·2-s + 1.67e7·4-s − 3.41e8·5-s − 4.08e10·7-s + 6.87e10·8-s − 1.39e12·10-s + 1.45e13·11-s + 8.78e13·13-s − 1.67e14·14-s + 2.81e14·16-s + 2.65e15·17-s − 1.39e16·19-s − 5.72e15·20-s + 5.94e16·22-s − 8.58e16·23-s − 1.81e17·25-s + 3.59e17·26-s − 6.85e17·28-s − 2.08e18·29-s + 2.66e18·31-s + 1.15e18·32-s + 1.08e19·34-s + 1.39e19·35-s − 5.13e19·37-s − 5.73e19·38-s − 2.34e19·40-s − 2.33e20·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.624·5-s − 1.11·7-s + 0.353·8-s − 0.441·10-s + 1.39·11-s + 1.04·13-s − 0.789·14-s + 1/4·16-s + 1.10·17-s − 1.45·19-s − 0.312·20-s + 0.985·22-s − 0.816·23-s − 0.609·25-s + 0.739·26-s − 0.558·28-s − 1.09·29-s + 0.607·31-s + 0.176·32-s + 0.781·34-s + 0.697·35-s − 1.28·37-s − 1.02·38-s − 0.220·40-s − 1.61·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(13)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{27}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{12} T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 13640214 p^{2} T + p^{25} T^{2} \) |
| 7 | \( 1 + 40882637368 T + p^{25} T^{2} \) |
| 11 | \( 1 - 119886135348 p^{2} T + p^{25} T^{2} \) |
| 13 | \( 1 - 87843989537006 T + p^{25} T^{2} \) |
| 17 | \( 1 - 156201521699214 p T + p^{25} T^{2} \) |
| 19 | \( 1 + 736811826531460 p T + p^{25} T^{2} \) |
| 23 | \( 1 + 3732729596697192 p T + p^{25} T^{2} \) |
| 29 | \( 1 + 2080230429601526910 T + p^{25} T^{2} \) |
| 31 | \( 1 - 2663532371302675232 T + p^{25} T^{2} \) |
| 37 | \( 1 + 51379607980315436218 T + p^{25} T^{2} \) |
| 41 | \( 1 + \)\(23\!\cdots\!22\)\( T + p^{25} T^{2} \) |
| 43 | \( 1 + 40133597094729613684 T + p^{25} T^{2} \) |
| 47 | \( 1 + \)\(27\!\cdots\!72\)\( T + p^{25} T^{2} \) |
| 53 | \( 1 + \)\(42\!\cdots\!26\)\( T + p^{25} T^{2} \) |
| 59 | \( 1 - \)\(83\!\cdots\!80\)\( T + p^{25} T^{2} \) |
| 61 | \( 1 - \)\(24\!\cdots\!62\)\( T + p^{25} T^{2} \) |
| 67 | \( 1 + \)\(12\!\cdots\!28\)\( T + p^{25} T^{2} \) |
| 71 | \( 1 - \)\(93\!\cdots\!88\)\( T + p^{25} T^{2} \) |
| 73 | \( 1 - \)\(40\!\cdots\!86\)\( T + p^{25} T^{2} \) |
| 79 | \( 1 + \)\(80\!\cdots\!80\)\( T + p^{25} T^{2} \) |
| 83 | \( 1 + \)\(89\!\cdots\!76\)\( T + p^{25} T^{2} \) |
| 89 | \( 1 + \)\(35\!\cdots\!90\)\( T + p^{25} T^{2} \) |
| 97 | \( 1 + \)\(86\!\cdots\!18\)\( T + p^{25} T^{2} \) |
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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
−12.47918060354774852292990912212, −11.52601097670495176333955062229, −10.01312318243511681098432771062, −8.477575743677921995434183435296, −6.83148953822533445768237621135, −5.92339853459057091927092969215, −4.00824070491925718110060710731, −3.43118583143186629531591111244, −1.59958114214265723002936471344, 0,
1.59958114214265723002936471344, 3.43118583143186629531591111244, 4.00824070491925718110060710731, 5.92339853459057091927092969215, 6.83148953822533445768237621135, 8.477575743677921995434183435296, 10.01312318243511681098432771062, 11.52601097670495176333955062229, 12.47918060354774852292990912212
