| L(s) = 1 | + 229.·2-s + 1.27e4·3-s − 7.82e4·4-s + 9.30e4·5-s + 2.92e6·6-s + 1.03e7·7-s − 4.80e7·8-s + 3.27e7·9-s + 2.13e7·10-s − 7.26e8·11-s − 9.96e8·12-s + 8.27e8·13-s + 2.37e9·14-s + 1.18e9·15-s − 7.87e8·16-s − 2.20e10·17-s + 7.52e9·18-s + 3.65e10·19-s − 7.28e9·20-s + 1.31e11·21-s − 1.66e11·22-s − 4.80e11·23-s − 6.12e11·24-s − 7.54e11·25-s + 1.90e11·26-s − 1.22e12·27-s − 8.09e11·28-s + ⋯ |
| L(s) = 1 | + 0.634·2-s + 1.11·3-s − 0.597·4-s + 0.106·5-s + 0.710·6-s + 0.677·7-s − 1.01·8-s + 0.253·9-s + 0.0675·10-s − 1.02·11-s − 0.668·12-s + 0.281·13-s + 0.430·14-s + 0.119·15-s − 0.0458·16-s − 0.764·17-s + 0.160·18-s + 0.493·19-s − 0.0636·20-s + 0.759·21-s − 0.648·22-s − 1.27·23-s − 1.13·24-s − 0.988·25-s + 0.178·26-s − 0.835·27-s − 0.404·28-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 | \( 1 - 229.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.27e4T + 1.29e8T^{2} \) |
| 5 | \( 1 - 9.30e4T + 7.62e11T^{2} \) |
| 7 | \( 1 - 1.03e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 7.26e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 8.27e8T + 8.65e18T^{2} \) |
| 17 | \( 1 + 2.20e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 3.65e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 4.80e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 2.08e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.08e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 7.30e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 2.95e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.79e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 4.29e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 7.94e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 1.58e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.15e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 2.33e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 4.97e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 2.38e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 4.82e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 8.92e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 1.11e17T + 5.95e33T^{2} \) |
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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
−13.21490576904014449321573775850, −11.67103661101186202193996261740, −9.941601389494375899067280296973, −8.676544196568731796975973016699, −7.85469801482059527693667308428, −5.76283596117951303481387487767, −4.47242211083933764016949992570, −3.24360966499525448664935436797, −2.01503827487184052576785438239, 0,
2.01503827487184052576785438239, 3.24360966499525448664935436797, 4.47242211083933764016949992570, 5.76283596117951303481387487767, 7.85469801482059527693667308428, 8.676544196568731796975973016699, 9.941601389494375899067280296973, 11.67103661101186202193996261740, 13.21490576904014449321573775850
