| L(s) = 1 | + 2-s + 4-s − 1.37·5-s + 7-s + 8-s − 1.37·10-s + 1.37·11-s + 3.37·13-s + 14-s + 16-s − 17-s + 2·19-s − 1.37·20-s + 1.37·22-s − 2.74·23-s − 3.11·25-s + 3.37·26-s + 28-s + 2.74·29-s + 8·31-s + 32-s − 34-s − 1.37·35-s + 0.627·37-s + 2·38-s − 1.37·40-s + 8.74·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.613·5-s + 0.377·7-s + 0.353·8-s − 0.433·10-s + 0.413·11-s + 0.935·13-s + 0.267·14-s + 0.250·16-s − 0.242·17-s + 0.458·19-s − 0.306·20-s + 0.292·22-s − 0.572·23-s − 0.623·25-s + 0.661·26-s + 0.188·28-s + 0.509·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.231·35-s + 0.103·37-s + 0.324·38-s − 0.216·40-s + 1.36·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2142 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.830038611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.830038611\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 1.37T + 5T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 3.37T + 13T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 0.627T + 37T^{2} \) |
| 41 | \( 1 - 8.74T + 41T^{2} \) |
| 43 | \( 1 - 0.627T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 3.37T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.627T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 4.11T + 89T^{2} \) |
| 97 | \( 1 - 0.116T + 97T^{2} \) |
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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
−8.988864246482126207065756767847, −8.107109593904361811401694465440, −7.61737109986948093293620337859, −6.53031044742852061454041527917, −5.99959187780054553544343073294, −4.93334225561571692906348937452, −4.15737911630307127048014490899, −3.50012150824820184902996288588, −2.35298961574445861834357933398, −1.05819833108159968113614247558,
1.05819833108159968113614247558, 2.35298961574445861834357933398, 3.50012150824820184902996288588, 4.15737911630307127048014490899, 4.93334225561571692906348937452, 5.99959187780054553544343073294, 6.53031044742852061454041527917, 7.61737109986948093293620337859, 8.107109593904361811401694465440, 8.988864246482126207065756767847
