| L(s) = 1 | − 2.46·2-s + 0.639·3-s + 4.09·4-s − 1.75·5-s − 1.58·6-s − 0.0776·7-s − 5.18·8-s − 2.59·9-s + 4.34·10-s − 0.622·11-s + 2.62·12-s − 2.61·13-s + 0.191·14-s − 1.12·15-s + 4.60·16-s − 0.276·17-s + 6.39·18-s − 6.23·19-s − 7.20·20-s − 0.0496·21-s + 1.53·22-s − 1.37·23-s − 3.31·24-s − 1.91·25-s + 6.46·26-s − 3.57·27-s − 0.318·28-s + ⋯ |
| L(s) = 1 | − 1.74·2-s + 0.369·3-s + 2.04·4-s − 0.786·5-s − 0.645·6-s − 0.0293·7-s − 1.83·8-s − 0.863·9-s + 1.37·10-s − 0.187·11-s + 0.757·12-s − 0.726·13-s + 0.0512·14-s − 0.290·15-s + 1.15·16-s − 0.0670·17-s + 1.50·18-s − 1.43·19-s − 1.61·20-s − 0.0108·21-s + 0.327·22-s − 0.286·23-s − 0.677·24-s − 0.382·25-s + 1.26·26-s − 0.688·27-s − 0.0601·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2326196438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2326196438\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 3 | \( 1 - 0.639T + 3T^{2} \) |
| 5 | \( 1 + 1.75T + 5T^{2} \) |
| 7 | \( 1 + 0.0776T + 7T^{2} \) |
| 11 | \( 1 + 0.622T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 0.276T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 1.37T + 23T^{2} \) |
| 29 | \( 1 + 1.18T + 29T^{2} \) |
| 31 | \( 1 - 0.419T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 - 6.47T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 + 0.474T + 47T^{2} \) |
| 53 | \( 1 + 3.44T + 53T^{2} \) |
| 59 | \( 1 - 3.79T + 59T^{2} \) |
| 61 | \( 1 + 1.11T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 0.468T + 71T^{2} \) |
| 73 | \( 1 - 8.13T + 73T^{2} \) |
| 79 | \( 1 - 6.79T + 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 8.61T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.258513592440152146209401653741, −8.096128875833721849886427934768, −7.33773707079432653383662490691, −6.60735258679835093041794958810, −5.78306334303558174071687890741, −4.60200153396241844221325364710, −3.55793036071728041996541756580, −2.57605890071586501049896870759, −1.87906876363855054437518151297, −0.33131415313551619239211024614,
0.33131415313551619239211024614, 1.87906876363855054437518151297, 2.57605890071586501049896870759, 3.55793036071728041996541756580, 4.60200153396241844221325364710, 5.78306334303558174071687890741, 6.60735258679835093041794958810, 7.33773707079432653383662490691, 8.096128875833721849886427934768, 8.258513592440152146209401653741
