| L(s) = 1 | + 0.743·2-s + 0.765·3-s − 7.44·4-s + 3.68·5-s + 0.569·6-s + 7·7-s − 11.4·8-s − 26.4·9-s + 2.73·10-s − 1.75·11-s − 5.70·12-s − 35.3·13-s + 5.20·14-s + 2.81·15-s + 51.0·16-s − 72.1·17-s − 19.6·18-s − 104.·19-s − 27.4·20-s + 5.35·21-s − 1.30·22-s − 23·23-s − 8.79·24-s − 111.·25-s − 26.3·26-s − 40.8·27-s − 52.1·28-s + ⋯ |
| L(s) = 1 | + 0.262·2-s + 0.147·3-s − 0.930·4-s + 0.329·5-s + 0.0387·6-s + 0.377·7-s − 0.507·8-s − 0.978·9-s + 0.0865·10-s − 0.0479·11-s − 0.137·12-s − 0.754·13-s + 0.0993·14-s + 0.0485·15-s + 0.797·16-s − 1.02·17-s − 0.257·18-s − 1.25·19-s − 0.306·20-s + 0.0556·21-s − 0.0126·22-s − 0.208·23-s − 0.0748·24-s − 0.891·25-s − 0.198·26-s − 0.291·27-s − 0.351·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 - 0.743T + 8T^{2} \) |
| 3 | \( 1 - 0.765T + 27T^{2} \) |
| 5 | \( 1 - 3.68T + 125T^{2} \) |
| 11 | \( 1 + 1.75T + 1.33e3T^{2} \) |
| 13 | \( 1 + 35.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 72.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 33.5T + 2.43e4T^{2} \) |
| 31 | \( 1 - 39.4T + 2.97e4T^{2} \) |
| 37 | \( 1 - 276.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 357.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 63.6T + 7.95e4T^{2} \) |
| 47 | \( 1 - 483.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 535.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 82.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 598.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 748.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 707.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 954.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 272.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 505.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 297.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 634.T + 9.12e5T^{2} \) |
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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
−12.04444728152393625829802964763, −10.94170508720013337795630826134, −9.728264854115653475146513547989, −8.816307196142727213606205174572, −7.976244860906441861846167481753, −6.29860698365964021975628556994, −5.18855693587603169022759703011, −4.08030774200908676244299570416, −2.39682505596709460683770567098, 0,
2.39682505596709460683770567098, 4.08030774200908676244299570416, 5.18855693587603169022759703011, 6.29860698365964021975628556994, 7.976244860906441861846167481753, 8.816307196142727213606205174572, 9.728264854115653475146513547989, 10.94170508720013337795630826134, 12.04444728152393625829802964763
