| L(s) = 1 | − 2-s − 0.548·3-s + 4-s − 5-s + 0.548·6-s − 1.86·7-s − 8-s − 2.69·9-s + 10-s + 0.761·11-s − 0.548·12-s − 4.22·13-s + 1.86·14-s + 0.548·15-s + 16-s + 1.77·17-s + 2.69·18-s − 7.51·19-s − 20-s + 1.02·21-s − 0.761·22-s + 2.16·23-s + 0.548·24-s + 25-s + 4.22·26-s + 3.12·27-s − 1.86·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.316·3-s + 0.5·4-s − 0.447·5-s + 0.224·6-s − 0.705·7-s − 0.353·8-s − 0.899·9-s + 0.316·10-s + 0.229·11-s − 0.158·12-s − 1.17·13-s + 0.498·14-s + 0.141·15-s + 0.250·16-s + 0.431·17-s + 0.636·18-s − 1.72·19-s − 0.223·20-s + 0.223·21-s − 0.162·22-s + 0.451·23-s + 0.112·24-s + 0.200·25-s + 0.827·26-s + 0.601·27-s − 0.352·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3109046209\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3109046209\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 0.548T + 3T^{2} \) |
| 7 | \( 1 + 1.86T + 7T^{2} \) |
| 11 | \( 1 - 0.761T + 11T^{2} \) |
| 13 | \( 1 + 4.22T + 13T^{2} \) |
| 17 | \( 1 - 1.77T + 17T^{2} \) |
| 19 | \( 1 + 7.51T + 19T^{2} \) |
| 23 | \( 1 - 2.16T + 23T^{2} \) |
| 29 | \( 1 + 1.35T + 29T^{2} \) |
| 31 | \( 1 - 2.74T + 31T^{2} \) |
| 37 | \( 1 + 8.22T + 37T^{2} \) |
| 41 | \( 1 + 11.9T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 3.51T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 6.48T + 61T^{2} \) |
| 67 | \( 1 - 7.99T + 67T^{2} \) |
| 71 | \( 1 + 4.51T + 71T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 13.9T + 79T^{2} \) |
| 83 | \( 1 + 0.0586T + 83T^{2} \) |
| 89 | \( 1 + 0.122T + 89T^{2} \) |
| 97 | \( 1 + 2.83T + 97T^{2} \) |
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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
−8.557076778452923054069065028066, −7.83905962955455113862772085195, −6.76669865309881385429202786334, −6.63971997707151408308533986044, −5.52479304561863075032363885022, −4.81558819994914014187441866035, −3.64897669252555946164734557405, −2.89512424211676761924850598521, −1.91841981022157040927835646102, −0.33924421108008865400279311085,
0.33924421108008865400279311085, 1.91841981022157040927835646102, 2.89512424211676761924850598521, 3.64897669252555946164734557405, 4.81558819994914014187441866035, 5.52479304561863075032363885022, 6.63971997707151408308533986044, 6.76669865309881385429202786334, 7.83905962955455113862772085195, 8.557076778452923054069065028066
