| L(s) = 1 | + 2·2-s − 9·3-s + 4·4-s − 20·5-s − 18·6-s + 2·7-s + 8·8-s + 54·9-s − 40·10-s − 52·11-s − 36·12-s + 43·13-s + 4·14-s + 180·15-s + 16·16-s − 50·17-s + 108·18-s − 74·19-s − 80·20-s − 18·21-s − 104·22-s − 23·23-s − 72·24-s + 275·25-s + 86·26-s − 243·27-s + 8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.78·5-s − 1.22·6-s + 0.107·7-s + 0.353·8-s + 2·9-s − 1.26·10-s − 1.42·11-s − 0.866·12-s + 0.917·13-s + 0.0763·14-s + 3.09·15-s + 1/4·16-s − 0.713·17-s + 1.41·18-s − 0.893·19-s − 0.894·20-s − 0.187·21-s − 1.00·22-s − 0.208·23-s − 0.612·24-s + 11/5·25-s + 0.648·26-s − 1.73·27-s + 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 + p^{2} T + p^{3} T^{2} \) |
| 5 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 52 T + p^{3} T^{2} \) |
| 13 | \( 1 - 43 T + p^{3} T^{2} \) |
| 17 | \( 1 + 50 T + p^{3} T^{2} \) |
| 19 | \( 1 + 74 T + p^{3} T^{2} \) |
| 29 | \( 1 + 7 T + p^{3} T^{2} \) |
| 31 | \( 1 + 273 T + p^{3} T^{2} \) |
| 37 | \( 1 + 4 T + p^{3} T^{2} \) |
| 41 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 43 | \( 1 + 152 T + p^{3} T^{2} \) |
| 47 | \( 1 - 75 T + p^{3} T^{2} \) |
| 53 | \( 1 - 86 T + p^{3} T^{2} \) |
| 59 | \( 1 + 444 T + p^{3} T^{2} \) |
| 61 | \( 1 - 262 T + p^{3} T^{2} \) |
| 67 | \( 1 - 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 21 T + p^{3} T^{2} \) |
| 73 | \( 1 - 681 T + p^{3} T^{2} \) |
| 79 | \( 1 - 426 T + p^{3} T^{2} \) |
| 83 | \( 1 - 902 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1272 T + p^{3} T^{2} \) |
| 97 | \( 1 + 342 T + p^{3} T^{2} \) |
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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
−15.17829693818564592395981637201, −13.04026521406439713260220637821, −12.29262848030218702304989807874, −11.06489307202725392941053699161, −10.89255348523744818745676957723, −8.004078591631948866550850672539, −6.72433686628161349578886524736, −5.23334813174777030696628125219, −4.04511014073209817450154864302, 0,
4.04511014073209817450154864302, 5.23334813174777030696628125219, 6.72433686628161349578886524736, 8.004078591631948866550850672539, 10.89255348523744818745676957723, 11.06489307202725392941053699161, 12.29262848030218702304989807874, 13.04026521406439713260220637821, 15.17829693818564592395981637201
