| L(s) = 1 | − 2-s − 2.45·3-s + 4-s + 0.358·5-s + 2.45·6-s − 1.91·7-s − 8-s + 3.03·9-s − 0.358·10-s + 5.12·11-s − 2.45·12-s + 3.67·13-s + 1.91·14-s − 0.880·15-s + 16-s + 3.57·17-s − 3.03·18-s − 7.77·19-s + 0.358·20-s + 4.69·21-s − 5.12·22-s − 0.723·23-s + 2.45·24-s − 4.87·25-s − 3.67·26-s − 0.0805·27-s − 1.91·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.41·3-s + 0.5·4-s + 0.160·5-s + 1.00·6-s − 0.722·7-s − 0.353·8-s + 1.01·9-s − 0.113·10-s + 1.54·11-s − 0.709·12-s + 1.01·13-s + 0.510·14-s − 0.227·15-s + 0.250·16-s + 0.867·17-s − 0.714·18-s − 1.78·19-s + 0.0801·20-s + 1.02·21-s − 1.09·22-s − 0.150·23-s + 0.501·24-s − 0.974·25-s − 0.721·26-s − 0.0155·27-s − 0.361·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8655210574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8655210574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 4001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 - 0.358T + 5T^{2} \) |
| 7 | \( 1 + 1.91T + 7T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 - 3.67T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 + 7.77T + 19T^{2} \) |
| 23 | \( 1 + 0.723T + 23T^{2} \) |
| 29 | \( 1 - 0.330T + 29T^{2} \) |
| 31 | \( 1 - 7.74T + 31T^{2} \) |
| 37 | \( 1 - 6.25T + 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 1.93T + 47T^{2} \) |
| 53 | \( 1 + 8.76T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 - 13.8T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 2.13T + 73T^{2} \) |
| 79 | \( 1 - 2.66T + 79T^{2} \) |
| 83 | \( 1 - 16.1T + 83T^{2} \) |
| 89 | \( 1 - 6.65T + 89T^{2} \) |
| 97 | \( 1 + 15.5T + 97T^{2} \) |
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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
−7.938841375674240757369536654935, −6.70267402031177690946554526555, −6.45053682327023722844842491973, −6.11942439810951969621995794628, −5.31429456816383496074633376886, −4.16794503920035373523184410642, −3.73739111946665980881820293019, −2.44784898617327320551977761398, −1.32281366959920603620880278052, −0.62434729523403802197540698415,
0.62434729523403802197540698415, 1.32281366959920603620880278052, 2.44784898617327320551977761398, 3.73739111946665980881820293019, 4.16794503920035373523184410642, 5.31429456816383496074633376886, 6.11942439810951969621995794628, 6.45053682327023722844842491973, 6.70267402031177690946554526555, 7.938841375674240757369536654935
