| L(s) = 1 | + 577.·2-s + 2.59e3·3-s + 2.01e5·4-s − 9.42e5·5-s + 1.49e6·6-s + 2.00e7·7-s + 4.08e7·8-s − 1.22e8·9-s − 5.43e8·10-s − 1.00e9·11-s + 5.24e8·12-s − 1.90e9·13-s + 1.15e10·14-s − 2.44e9·15-s − 2.88e9·16-s + 1.91e10·17-s − 7.06e10·18-s − 5.50e10·19-s − 1.90e11·20-s + 5.20e10·21-s − 5.77e11·22-s + 2.88e11·23-s + 1.06e11·24-s + 1.24e11·25-s − 1.10e12·26-s − 6.53e11·27-s + 4.04e12·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.228·3-s + 1.54·4-s − 1.07·5-s + 0.364·6-s + 1.31·7-s + 0.860·8-s − 0.947·9-s − 1.71·10-s − 1.40·11-s + 0.351·12-s − 0.648·13-s + 2.09·14-s − 0.246·15-s − 0.168·16-s + 0.665·17-s − 1.51·18-s − 0.744·19-s − 1.66·20-s + 0.300·21-s − 2.24·22-s + 0.767·23-s + 0.196·24-s + 0.163·25-s − 1.03·26-s − 0.445·27-s + 2.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 29 | \( 1 + 5.00e11T \) |
| good | 2 | \( 1 - 577.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 2.59e3T + 1.29e8T^{2} \) |
| 5 | \( 1 + 9.42e5T + 7.62e11T^{2} \) |
| 7 | \( 1 - 2.00e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 1.00e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.90e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.91e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 5.50e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.88e11T + 1.41e23T^{2} \) |
| 31 | \( 1 + 2.87e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 1.13e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.16e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 6.95e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.18e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 1.64e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 2.08e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.72e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.38e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 3.24e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 3.05e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.53e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.64e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 1.19e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.29e17T + 5.95e33T^{2} \) |
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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
−12.72343794086110583028205422468, −11.69421383083745045546968473223, −10.88496284650567730201537076115, −8.377371725458687148161079991114, −7.42373703346381606291508441040, −5.47941005352803985492904026002, −4.70947374991638277609708079903, −3.38404907887022374420900689291, −2.25208096371400810770982901328, 0,
2.25208096371400810770982901328, 3.38404907887022374420900689291, 4.70947374991638277609708079903, 5.47941005352803985492904026002, 7.42373703346381606291508441040, 8.377371725458687148161079991114, 10.88496284650567730201537076115, 11.69421383083745045546968473223, 12.72343794086110583028205422468
