| L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s − 0.618·6-s − 7-s − 2.23·8-s + 9-s + 11-s + 1.61·12-s − 3.47·13-s − 0.618·14-s + 1.85·16-s − 5.23·17-s + 0.618·18-s − 6.70·19-s + 21-s + 0.618·22-s − 5.70·23-s + 2.23·24-s − 2.14·26-s − 27-s + 1.61·28-s + 5·29-s − 5.23·31-s + 5.61·32-s − 33-s − 3.23·34-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s − 0.252·6-s − 0.377·7-s − 0.790·8-s + 0.333·9-s + 0.301·11-s + 0.467·12-s − 0.962·13-s − 0.165·14-s + 0.463·16-s − 1.26·17-s + 0.145·18-s − 1.53·19-s + 0.218·21-s + 0.131·22-s − 1.19·23-s + 0.456·24-s − 0.420·26-s − 0.192·27-s + 0.305·28-s + 0.928·29-s − 0.940·31-s + 0.993·32-s − 0.174·33-s − 0.554·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5217340343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5217340343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 13 | \( 1 + 3.47T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 6.70T + 19T^{2} \) |
| 23 | \( 1 + 5.70T + 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 + 2.47T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 + 0.236T + 47T^{2} \) |
| 53 | \( 1 - 12.1T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 + 2.47T + 71T^{2} \) |
| 73 | \( 1 + 4.52T + 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 + 6.76T + 83T^{2} \) |
| 89 | \( 1 + 4.47T + 89T^{2} \) |
| 97 | \( 1 + 9.70T + 97T^{2} \) |
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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
−8.274275376760460704146746634063, −7.21593339437810307273754383012, −6.50404685730716332838904591580, −5.97195037148174805710929971665, −5.16247113113174703653159966572, −4.29166831624316656138408870930, −4.13605423404094361011755694007, −2.87033988836941557577167206047, −1.93329324593234579873764040439, −0.35356167827866478505652865529,
0.35356167827866478505652865529, 1.93329324593234579873764040439, 2.87033988836941557577167206047, 4.13605423404094361011755694007, 4.29166831624316656138408870930, 5.16247113113174703653159966572, 5.97195037148174805710929971665, 6.50404685730716332838904591580, 7.21593339437810307273754383012, 8.274275376760460704146746634063
