| L(s) = 1 | − 0.319·2-s + 3-s − 1.89·4-s − 2.81·5-s − 0.319·6-s − 2.62·7-s + 1.24·8-s + 9-s + 0.897·10-s − 1.49·11-s − 1.89·12-s + 1.38·13-s + 0.839·14-s − 2.81·15-s + 3.39·16-s − 17-s − 0.319·18-s + 5.81·19-s + 5.33·20-s − 2.62·21-s + 0.477·22-s + 2.82·23-s + 1.24·24-s + 2.89·25-s − 0.442·26-s + 27-s + 4.98·28-s + ⋯ |
| L(s) = 1 | − 0.225·2-s + 0.577·3-s − 0.949·4-s − 1.25·5-s − 0.130·6-s − 0.993·7-s + 0.440·8-s + 0.333·9-s + 0.283·10-s − 0.451·11-s − 0.547·12-s + 0.383·13-s + 0.224·14-s − 0.725·15-s + 0.849·16-s − 0.242·17-s − 0.0752·18-s + 1.33·19-s + 1.19·20-s − 0.573·21-s + 0.101·22-s + 0.588·23-s + 0.254·24-s + 0.579·25-s − 0.0866·26-s + 0.192·27-s + 0.942·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.319T + 2T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 + 2.62T + 7T^{2} \) |
| 11 | \( 1 + 1.49T + 11T^{2} \) |
| 13 | \( 1 - 1.38T + 13T^{2} \) |
| 19 | \( 1 - 5.81T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 0.415T + 29T^{2} \) |
| 31 | \( 1 + 0.794T + 31T^{2} \) |
| 37 | \( 1 - 3.13T + 37T^{2} \) |
| 41 | \( 1 - 7.83T + 41T^{2} \) |
| 43 | \( 1 + 4.83T + 43T^{2} \) |
| 47 | \( 1 - 6.29T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 2.97T + 67T^{2} \) |
| 71 | \( 1 + 9.04T + 71T^{2} \) |
| 73 | \( 1 - 1.42T + 73T^{2} \) |
| 83 | \( 1 + 1.32T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−8.089157676372583243582720861037, −7.57738843791517659041314788087, −6.88634424913950625593332718981, −5.79369860373260373676370380904, −4.89065602750309278250592837053, −4.08383252129581740685472442331, −3.47067615942680060997110538073, −2.81256223281406515662651852656, −1.09325542139064300993100098093, 0,
1.09325542139064300993100098093, 2.81256223281406515662651852656, 3.47067615942680060997110538073, 4.08383252129581740685472442331, 4.89065602750309278250592837053, 5.79369860373260373676370380904, 6.88634424913950625593332718981, 7.57738843791517659041314788087, 8.089157676372583243582720861037
