| L(s) = 1 | + 421.·2-s + 1.49e4·3-s + 4.64e4·4-s − 5.71e4·5-s + 6.29e6·6-s − 1.89e7·7-s − 3.56e7·8-s + 9.37e7·9-s − 2.40e7·10-s + 3.80e8·11-s + 6.93e8·12-s − 4.45e9·13-s − 7.99e9·14-s − 8.53e8·15-s − 2.11e10·16-s + 1.79e10·17-s + 3.95e10·18-s − 4.97e10·19-s − 2.65e9·20-s − 2.83e11·21-s + 1.60e11·22-s + 1.61e10·23-s − 5.32e11·24-s − 7.59e11·25-s − 1.87e12·26-s − 5.27e11·27-s − 8.81e11·28-s + ⋯ |
| L(s) = 1 | + 1.16·2-s + 1.31·3-s + 0.354·4-s − 0.0654·5-s + 1.52·6-s − 1.24·7-s − 0.751·8-s + 0.726·9-s − 0.0761·10-s + 0.534·11-s + 0.465·12-s − 1.51·13-s − 1.44·14-s − 0.0860·15-s − 1.22·16-s + 0.623·17-s + 0.845·18-s − 0.672·19-s − 0.0232·20-s − 1.63·21-s + 0.622·22-s + 0.0431·23-s − 0.987·24-s − 0.995·25-s − 1.76·26-s − 0.359·27-s − 0.440·28-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 | \( 1 - 421.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.49e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 5.71e4T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.89e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 3.80e8T + 5.05e17T^{2} \) |
| 13 | \( 1 + 4.45e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.79e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.97e10T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.61e10T + 1.41e23T^{2} \) |
| 31 | \( 1 - 4.17e11T + 2.25e25T^{2} \) |
| 37 | \( 1 + 3.42e12T + 4.56e26T^{2} \) |
| 41 | \( 1 + 2.02e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 1.29e14T + 5.87e27T^{2} \) |
| 47 | \( 1 - 2.32e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 2.54e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 3.95e14T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.15e13T + 2.24e30T^{2} \) |
| 67 | \( 1 - 6.48e13T + 1.10e31T^{2} \) |
| 71 | \( 1 + 7.12e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 9.52e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 1.32e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.13e16T + 4.21e32T^{2} \) |
| 89 | \( 1 - 1.86e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.05e17T + 5.95e33T^{2} \) |
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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
−12.93176118929594426693247894226, −12.08574095575043716203147007319, −9.830504472727762268284710023189, −8.993527660592278853959797640411, −7.36492957116110253635498206966, −5.92309022621292554830151620953, −4.25040775281115669571362304333, −3.26342659541246234904990461927, −2.37543285912728302907473128417, 0,
2.37543285912728302907473128417, 3.26342659541246234904990461927, 4.25040775281115669571362304333, 5.92309022621292554830151620953, 7.36492957116110253635498206966, 8.993527660592278853959797640411, 9.830504472727762268284710023189, 12.08574095575043716203147007319, 12.93176118929594426693247894226
