| L(s) = 1 | − 4.09e3·2-s + 9.79e4·3-s + 1.67e7·4-s + 3.41e8·5-s − 4.01e8·6-s − 4.08e10·7-s − 6.87e10·8-s − 8.37e11·9-s − 1.39e12·10-s − 1.45e13·11-s + 1.64e12·12-s + 8.78e13·13-s + 1.67e14·14-s + 3.34e13·15-s + 2.81e14·16-s − 2.65e15·17-s + 3.43e15·18-s − 1.39e16·19-s + 5.72e15·20-s − 4.00e15·21-s + 5.94e16·22-s + 8.58e16·23-s − 6.73e15·24-s − 1.81e17·25-s − 3.59e17·26-s − 1.65e17·27-s − 6.85e17·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.106·3-s + 1/2·4-s + 0.624·5-s − 0.0752·6-s − 1.11·7-s − 0.353·8-s − 0.988·9-s − 0.441·10-s − 1.39·11-s + 0.0532·12-s + 1.04·13-s + 0.789·14-s + 0.0664·15-s + 1/4·16-s − 1.10·17-s + 0.699·18-s − 1.45·19-s + 0.312·20-s − 0.118·21-s + 0.985·22-s + 0.816·23-s − 0.0376·24-s − 0.609·25-s − 0.739·26-s − 0.211·27-s − 0.558·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(26-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+25/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{12} T \) |
| good | 3 | \( 1 - 3628 p^{3} T + p^{25} T^{2} \) |
| 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
−21.03277881420194241599822402721, −19.30355055652015442141878442171, −17.57727726046822985350139551973, −15.79503871325042898732170886817, −13.25081672067766652440036689191, −10.60306246663859647606790956080, −8.726734908863575117706841522358, −6.19626757338253459622011448415, −2.62186641729518610637434996072, 0,
2.62186641729518610637434996072, 6.19626757338253459622011448415, 8.726734908863575117706841522358, 10.60306246663859647606790956080, 13.25081672067766652440036689191, 15.79503871325042898732170886817, 17.57727726046822985350139551973, 19.30355055652015442141878442171, 21.03277881420194241599822402721
