| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 3·9-s − 2·13-s − 14-s + 16-s + 3·18-s − 4·19-s − 8·23-s + 2·26-s + 28-s + 8·29-s + 8·31-s − 32-s − 3·36-s + 8·37-s + 4·38-s + 8·41-s − 4·43-s + 8·46-s + 8·47-s + 49-s − 2·52-s − 10·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 0.707·18-s − 0.917·19-s − 1.66·23-s + 0.392·26-s + 0.188·28-s + 1.48·29-s + 1.43·31-s − 0.176·32-s − 1/2·36-s + 1.31·37-s + 0.648·38-s + 1.24·41-s − 0.609·43-s + 1.17·46-s + 1.16·47-s + 1/7·49-s − 0.277·52-s − 1.37·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.214814737\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.214814737\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.96209352742756, −13.26432539371719, −12.59627977469835, −12.15912692244935, −11.67292890262284, −11.38555891063054, −10.68967984733096, −10.27115604766155, −9.841971745046216, −9.299855369414348, −8.576329752495305, −8.368753821989011, −7.873957350613708, −7.406324813162093, −6.593557894064172, −6.133036500759173, −5.862532146894507, −4.965574901255822, −4.474043685554152, −3.906218615836130, −2.982281623044004, −2.483027917723883, −2.096843838972004, −1.088569535667773, −0.4277654479652082,
0.4277654479652082, 1.088569535667773, 2.096843838972004, 2.483027917723883, 2.982281623044004, 3.906218615836130, 4.474043685554152, 4.965574901255822, 5.862532146894507, 6.133036500759173, 6.593557894064172, 7.406324813162093, 7.873957350613708, 8.368753821989011, 8.576329752495305, 9.299855369414348, 9.841971745046216, 10.27115604766155, 10.68967984733096, 11.38555891063054, 11.67292890262284, 12.15912692244935, 12.59627977469835, 13.26432539371719, 13.96209352742756
