| L(s) = 1 | + 2-s − 1.96·3-s + 4-s − 0.579·5-s − 1.96·6-s − 3.04·7-s + 8-s + 0.871·9-s − 0.579·10-s − 2.35·11-s − 1.96·12-s + 0.806·13-s − 3.04·14-s + 1.13·15-s + 16-s − 2.04·17-s + 0.871·18-s − 5.84·19-s − 0.579·20-s + 5.99·21-s − 2.35·22-s − 0.348·23-s − 1.96·24-s − 4.66·25-s + 0.806·26-s + 4.18·27-s − 3.04·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.13·3-s + 0.5·4-s − 0.259·5-s − 0.803·6-s − 1.15·7-s + 0.353·8-s + 0.290·9-s − 0.183·10-s − 0.709·11-s − 0.568·12-s + 0.223·13-s − 0.814·14-s + 0.294·15-s + 0.250·16-s − 0.495·17-s + 0.205·18-s − 1.34·19-s − 0.129·20-s + 1.30·21-s − 0.501·22-s − 0.0726·23-s − 0.401·24-s − 0.932·25-s + 0.158·26-s + 0.805·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8543598481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8543598481\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
| good | 3 | \( 1 + 1.96T + 3T^{2} \) |
| 5 | \( 1 + 0.579T + 5T^{2} \) |
| 7 | \( 1 + 3.04T + 7T^{2} \) |
| 11 | \( 1 + 2.35T + 11T^{2} \) |
| 13 | \( 1 - 0.806T + 13T^{2} \) |
| 17 | \( 1 + 2.04T + 17T^{2} \) |
| 19 | \( 1 + 5.84T + 19T^{2} \) |
| 23 | \( 1 + 0.348T + 23T^{2} \) |
| 29 | \( 1 - 0.290T + 29T^{2} \) |
| 31 | \( 1 - 2.67T + 31T^{2} \) |
| 37 | \( 1 - 1.58T + 37T^{2} \) |
| 41 | \( 1 + 6.23T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 3.85T + 59T^{2} \) |
| 61 | \( 1 + 3.53T + 61T^{2} \) |
| 67 | \( 1 + 1.09T + 67T^{2} \) |
| 71 | \( 1 - 8.65T + 71T^{2} \) |
| 73 | \( 1 - 0.458T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 7.13T + 83T^{2} \) |
| 89 | \( 1 - 8.36T + 89T^{2} \) |
| 97 | \( 1 + 6.79T + 97T^{2} \) |
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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
−8.324732550306782361155951443449, −7.50680872477442710157413961506, −6.52940930330318055979375015548, −6.26641834240724071302694353178, −5.57488816051578607228612793082, −4.72032349465862856175474531486, −4.01330281027176130279099179862, −3.07613377077819916409623796640, −2.15793660366559086669091830092, −0.47304532628871365741299593335,
0.47304532628871365741299593335, 2.15793660366559086669091830092, 3.07613377077819916409623796640, 4.01330281027176130279099179862, 4.72032349465862856175474531486, 5.57488816051578607228612793082, 6.26641834240724071302694353178, 6.52940930330318055979375015548, 7.50680872477442710157413961506, 8.324732550306782361155951443449
