| L(s) = 1 | − 2.75·2-s − 3-s + 5.58·4-s − 0.0622·5-s + 2.75·6-s + 7-s − 9.88·8-s + 9-s + 0.171·10-s + 1.07·11-s − 5.58·12-s + 6.04·13-s − 2.75·14-s + 0.0622·15-s + 16.0·16-s − 0.508·17-s − 2.75·18-s − 4.80·19-s − 0.347·20-s − 21-s − 2.95·22-s − 4.88·23-s + 9.88·24-s − 4.99·25-s − 16.6·26-s − 27-s + 5.58·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s − 0.577·3-s + 2.79·4-s − 0.0278·5-s + 1.12·6-s + 0.377·7-s − 3.49·8-s + 0.333·9-s + 0.0541·10-s + 0.323·11-s − 1.61·12-s + 1.67·13-s − 0.736·14-s + 0.0160·15-s + 4.01·16-s − 0.123·17-s − 0.649·18-s − 1.10·19-s − 0.0777·20-s − 0.218·21-s − 0.631·22-s − 1.01·23-s + 2.01·24-s − 0.999·25-s − 3.26·26-s − 0.192·27-s + 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5081775876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5081775876\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 383 | \( 1 + T \) |
| good | 2 | \( 1 + 2.75T + 2T^{2} \) |
| 5 | \( 1 + 0.0622T + 5T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 6.04T + 13T^{2} \) |
| 17 | \( 1 + 0.508T + 17T^{2} \) |
| 19 | \( 1 + 4.80T + 19T^{2} \) |
| 23 | \( 1 + 4.88T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 + 8.60T + 31T^{2} \) |
| 37 | \( 1 + 1.69T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 - 1.99T + 43T^{2} \) |
| 47 | \( 1 + 5.71T + 47T^{2} \) |
| 53 | \( 1 - 1.01T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 6.50T + 61T^{2} \) |
| 67 | \( 1 + 9.27T + 67T^{2} \) |
| 71 | \( 1 + 4.83T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 7.77T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.969518155974313574340002380155, −7.36149141857737934483141792908, −6.45595793512054506040934297036, −6.21158127914161553918505596039, −5.44319750187429716533065680106, −4.08441342764880326481336368428, −3.36309435225920479436078226278, −1.96165863185770072563635667885, −1.64705017551008533467998453332, −0.49395348131472030228997590524,
0.49395348131472030228997590524, 1.64705017551008533467998453332, 1.96165863185770072563635667885, 3.36309435225920479436078226278, 4.08441342764880326481336368428, 5.44319750187429716533065680106, 6.21158127914161553918505596039, 6.45595793512054506040934297036, 7.36149141857737934483141792908, 7.969518155974313574340002380155
