| L(s) = 1 | − 87.1·2-s + 1.18e4·3-s − 1.23e5·4-s − 5.09e5·5-s − 1.03e6·6-s − 8.46e6·7-s + 2.21e7·8-s + 1.22e7·9-s + 4.44e7·10-s + 1.21e9·11-s − 1.46e9·12-s + 3.67e9·13-s + 7.37e8·14-s − 6.06e9·15-s + 1.42e10·16-s + 1.11e9·17-s − 1.06e9·18-s − 4.26e10·19-s + 6.29e10·20-s − 1.00e11·21-s − 1.05e11·22-s − 2.90e11·23-s + 2.63e11·24-s − 5.02e11·25-s − 3.19e11·26-s − 1.38e12·27-s + 1.04e12·28-s + ⋯ |
| L(s) = 1 | − 0.240·2-s + 1.04·3-s − 0.942·4-s − 0.583·5-s − 0.251·6-s − 0.555·7-s + 0.467·8-s + 0.0950·9-s + 0.140·10-s + 1.70·11-s − 0.985·12-s + 1.24·13-s + 0.133·14-s − 0.610·15-s + 0.829·16-s + 0.0389·17-s − 0.0228·18-s − 0.576·19-s + 0.549·20-s − 0.580·21-s − 0.409·22-s − 0.772·23-s + 0.489·24-s − 0.659·25-s − 0.300·26-s − 0.946·27-s + 0.522·28-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 | \( 1 + 87.1T + 1.31e5T^{2} \) |
| 3 | \( 1 - 1.18e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 5.09e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 8.46e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.21e9T + 5.05e17T^{2} \) |
| 13 | \( 1 - 3.67e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.11e9T + 8.27e20T^{2} \) |
| 19 | \( 1 + 4.26e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.90e11T + 1.41e23T^{2} \) |
| 31 | \( 1 + 2.87e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 1.86e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 3.22e13T + 2.61e27T^{2} \) |
| 43 | \( 1 + 1.08e14T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.67e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 8.32e14T + 2.05e29T^{2} \) |
| 59 | \( 1 + 1.16e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 2.79e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 9.97e14T + 1.10e31T^{2} \) |
| 71 | \( 1 + 6.23e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 2.78e15T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.59e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 2.95e15T + 4.21e32T^{2} \) |
| 89 | \( 1 - 3.77e16T + 1.37e33T^{2} \) |
| 97 | \( 1 + 1.78e16T + 5.95e33T^{2} \) |
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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
−12.95975980733952720945009791243, −11.45062211607339664275259907663, −9.660525882970080008812550309247, −8.842126268223989056914955700701, −7.974850009588888501355706749989, −6.22127626673715083027290873658, −4.07061475098164204161614091577, −3.48813699640191913998293500098, −1.50065562013251599247065299863, 0,
1.50065562013251599247065299863, 3.48813699640191913998293500098, 4.07061475098164204161614091577, 6.22127626673715083027290873658, 7.974850009588888501355706749989, 8.842126268223989056914955700701, 9.660525882970080008812550309247, 11.45062211607339664275259907663, 12.95975980733952720945009791243
