| L(s) = 1 | − 483.·2-s + 1.98e4·3-s + 1.02e5·4-s − 3.00e5·5-s − 9.60e6·6-s − 1.05e7·7-s + 1.38e7·8-s + 2.65e8·9-s + 1.44e8·10-s − 3.33e8·11-s + 2.03e9·12-s + 2.39e9·13-s + 5.08e9·14-s − 5.96e9·15-s − 2.01e10·16-s − 3.71e10·17-s − 1.28e11·18-s − 5.58e9·19-s − 3.07e10·20-s − 2.09e11·21-s + 1.61e11·22-s + 2.74e11·23-s + 2.74e11·24-s − 6.72e11·25-s − 1.15e12·26-s + 2.71e12·27-s − 1.07e12·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 1.74·3-s + 0.782·4-s − 0.343·5-s − 2.33·6-s − 0.689·7-s + 0.290·8-s + 2.05·9-s + 0.458·10-s − 0.469·11-s + 1.36·12-s + 0.812·13-s + 0.921·14-s − 0.600·15-s − 1.17·16-s − 1.29·17-s − 2.74·18-s − 0.0754·19-s − 0.268·20-s − 1.20·21-s + 0.626·22-s + 0.729·23-s + 0.508·24-s − 0.882·25-s − 1.08·26-s + 1.84·27-s − 0.539·28-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 | \( 1 + 483.T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.98e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 3.00e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 1.05e7T + 2.32e14T^{2} \) |
| 11 | \( 1 + 3.33e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 2.39e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 3.71e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 5.58e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 2.74e11T + 1.41e23T^{2} \) |
| 31 | \( 1 - 1.59e12T + 2.25e25T^{2} \) |
| 37 | \( 1 + 2.65e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 5.75e12T + 2.61e27T^{2} \) |
| 43 | \( 1 - 9.28e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 3.14e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 5.46e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.21e15T + 1.27e30T^{2} \) |
| 61 | \( 1 + 2.14e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 5.99e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.60e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 6.06e14T + 4.74e31T^{2} \) |
| 79 | \( 1 - 3.90e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.77e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 5.40e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 2.26e16T + 5.95e33T^{2} \) |
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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
−13.02091077274059796480334246737, −10.86966424153001976084170660877, −9.632161953879354823829348992604, −8.828602411456487301740364103112, −8.015621984182079154165999610213, −6.90270248435348434549117940313, −4.11651764947175167517855209021, −2.80070899004159457013546251202, −1.58633064710683576484611272924, 0,
1.58633064710683576484611272924, 2.80070899004159457013546251202, 4.11651764947175167517855209021, 6.90270248435348434549117940313, 8.015621984182079154165999610213, 8.828602411456487301740364103112, 9.632161953879354823829348992604, 10.86966424153001976084170660877, 13.02091077274059796480334246737
