| L(s) = 1 | + 2-s − 3.04·3-s + 4-s − 0.106·5-s − 3.04·6-s − 0.613·7-s + 8-s + 6.29·9-s − 0.106·10-s + 4.91·11-s − 3.04·12-s − 1.74·13-s − 0.613·14-s + 0.325·15-s + 16-s + 0.753·17-s + 6.29·18-s + 0.918·19-s − 0.106·20-s + 1.87·21-s + 4.91·22-s − 3.35·23-s − 3.04·24-s − 4.98·25-s − 1.74·26-s − 10.0·27-s − 0.613·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.76·3-s + 0.5·4-s − 0.0477·5-s − 1.24·6-s − 0.232·7-s + 0.353·8-s + 2.09·9-s − 0.0337·10-s + 1.48·11-s − 0.880·12-s − 0.483·13-s − 0.164·14-s + 0.0839·15-s + 0.250·16-s + 0.182·17-s + 1.48·18-s + 0.210·19-s − 0.0238·20-s + 0.408·21-s + 1.04·22-s − 0.700·23-s − 0.622·24-s − 0.997·25-s − 0.341·26-s − 1.93·27-s − 0.116·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.04T + 3T^{2} \) |
| 5 | \( 1 + 0.106T + 5T^{2} \) |
| 7 | \( 1 + 0.613T + 7T^{2} \) |
| 11 | \( 1 - 4.91T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.753T + 17T^{2} \) |
| 19 | \( 1 - 0.918T + 19T^{2} \) |
| 23 | \( 1 + 3.35T + 23T^{2} \) |
| 29 | \( 1 - 3.52T + 29T^{2} \) |
| 31 | \( 1 + 6.13T + 31T^{2} \) |
| 37 | \( 1 + 1.78T + 37T^{2} \) |
| 41 | \( 1 + 9.94T + 41T^{2} \) |
| 43 | \( 1 - 5.22T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 - 8.55T + 59T^{2} \) |
| 61 | \( 1 + 3.22T + 61T^{2} \) |
| 67 | \( 1 + 4.18T + 67T^{2} \) |
| 71 | \( 1 + 0.630T + 71T^{2} \) |
| 73 | \( 1 + 0.360T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 + 2.30T + 97T^{2} \) |
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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
−7.13731209469954236993970575713, −6.45997919747577840995502103024, −6.20015051143019931783091136223, −5.34377135376734460860416406484, −4.87349467302301645341708443399, −4.00591555012798511077906712325, −3.52432728347690741441070956572, −2.03907651895477965033810940874, −1.20640200799872019834787562197, 0,
1.20640200799872019834787562197, 2.03907651895477965033810940874, 3.52432728347690741441070956572, 4.00591555012798511077906712325, 4.87349467302301645341708443399, 5.34377135376734460860416406484, 6.20015051143019931783091136223, 6.45997919747577840995502103024, 7.13731209469954236993970575713
