| L(s) = 1 | − 2.44·2-s − 1.15·3-s + 3.98·4-s − 3.94·5-s + 2.82·6-s + 3.67·7-s − 4.84·8-s − 1.66·9-s + 9.64·10-s − 4.34·11-s − 4.60·12-s + 2.56·13-s − 8.99·14-s + 4.55·15-s + 3.88·16-s − 1.43·17-s + 4.06·18-s − 3.57·19-s − 15.6·20-s − 4.25·21-s + 10.6·22-s + 7.56·23-s + 5.59·24-s + 10.5·25-s − 6.26·26-s + 5.39·27-s + 14.6·28-s + ⋯ |
| L(s) = 1 | − 1.72·2-s − 0.667·3-s + 1.99·4-s − 1.76·5-s + 1.15·6-s + 1.38·7-s − 1.71·8-s − 0.554·9-s + 3.04·10-s − 1.30·11-s − 1.32·12-s + 0.710·13-s − 2.40·14-s + 1.17·15-s + 0.970·16-s − 0.348·17-s + 0.958·18-s − 0.820·19-s − 3.50·20-s − 0.927·21-s + 2.26·22-s + 1.57·23-s + 1.14·24-s + 2.11·25-s − 1.22·26-s + 1.03·27-s + 2.76·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 4003 | \( 1+O(T) \) |
| good | 2 | \( 1 + 2.44T + 2T^{2} \) |
| 3 | \( 1 + 1.15T + 3T^{2} \) |
| 5 | \( 1 + 3.94T + 5T^{2} \) |
| 7 | \( 1 - 3.67T + 7T^{2} \) |
| 11 | \( 1 + 4.34T + 11T^{2} \) |
| 13 | \( 1 - 2.56T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 - 7.56T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 3.66T + 31T^{2} \) |
| 37 | \( 1 + 0.492T + 37T^{2} \) |
| 41 | \( 1 - 6.34T + 41T^{2} \) |
| 43 | \( 1 - 2.05T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 + 3.51T + 53T^{2} \) |
| 59 | \( 1 + 8.67T + 59T^{2} \) |
| 61 | \( 1 - 7.65T + 61T^{2} \) |
| 67 | \( 1 - 0.0212T + 67T^{2} \) |
| 71 | \( 1 + 0.157T + 71T^{2} \) |
| 73 | \( 1 + 13.0T + 73T^{2} \) |
| 79 | \( 1 + 1.22T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.174440312618130301562891521126, −7.49286622850782523673840815610, −7.23407096895901086419883508373, −6.03045290103355206507756572475, −5.11735990794833256358555002686, −4.36935195705466362703011992692, −3.18473110041699088216117915366, −2.10527444777212806282236774551, −0.860308406814733390804597485310, 0,
0.860308406814733390804597485310, 2.10527444777212806282236774551, 3.18473110041699088216117915366, 4.36935195705466362703011992692, 5.11735990794833256358555002686, 6.03045290103355206507756572475, 7.23407096895901086419883508373, 7.49286622850782523673840815610, 8.174440312618130301562891521126
