L(s) = 1 | + 2-s − 2.44·3-s + 4-s + 5-s − 2.44·6-s + 2.12·7-s + 8-s + 2.99·9-s + 10-s + 11-s − 2.44·12-s − 1.52·13-s + 2.12·14-s − 2.44·15-s + 16-s − 2.52·17-s + 2.99·18-s − 3.04·19-s + 20-s − 5.19·21-s + 22-s + 0.505·23-s − 2.44·24-s + 25-s − 1.52·26-s + 0.0159·27-s + 2.12·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.447·5-s − 0.999·6-s + 0.802·7-s + 0.353·8-s + 0.997·9-s + 0.316·10-s + 0.301·11-s − 0.706·12-s − 0.421·13-s + 0.567·14-s − 0.632·15-s + 0.250·16-s − 0.611·17-s + 0.705·18-s − 0.698·19-s + 0.223·20-s − 1.13·21-s + 0.213·22-s + 0.105·23-s − 0.499·24-s + 0.200·25-s − 0.298·26-s + 0.00307·27-s + 0.401·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.180697176\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.180697176\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 3 | \( 1 + 2.44T + 3T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 13 | \( 1 + 1.52T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 + 3.04T + 19T^{2} \) |
| 23 | \( 1 - 0.505T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 - 1.35T + 37T^{2} \) |
| 41 | \( 1 - 8.35T + 41T^{2} \) |
| 43 | \( 1 + 0.0405T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2.96T + 61T^{2} \) |
| 67 | \( 1 - 0.982T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 79 | \( 1 + 10.8T + 79T^{2} \) |
| 83 | \( 1 - 2.82T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−7.52599324220581285575369053909, −6.87811886086775657364326363432, −6.29832024493918434167772847178, −5.62211321050149675461126296019, −5.15950520570620596226332529840, −4.48588343714123746674982439179, −3.86622545290010691372246719294, −2.53410271371493813221875399913, −1.80426224647879737016291594180, −0.71006660830206080625096499446,
0.71006660830206080625096499446, 1.80426224647879737016291594180, 2.53410271371493813221875399913, 3.86622545290010691372246719294, 4.48588343714123746674982439179, 5.15950520570620596226332529840, 5.62211321050149675461126296019, 6.29832024493918434167772847178, 6.87811886086775657364326363432, 7.52599324220581285575369053909