| L(s) = 1 | + 2-s + 2.02·3-s + 4-s − 3.32·5-s + 2.02·6-s − 3.02·7-s + 8-s + 1.09·9-s − 3.32·10-s − 1.06·11-s + 2.02·12-s − 0.496·13-s − 3.02·14-s − 6.73·15-s + 16-s + 4.23·17-s + 1.09·18-s + 1.46·19-s − 3.32·20-s − 6.12·21-s − 1.06·22-s + 8.84·23-s + 2.02·24-s + 6.04·25-s − 0.496·26-s − 3.84·27-s − 3.02·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.16·3-s + 0.5·4-s − 1.48·5-s + 0.826·6-s − 1.14·7-s + 0.353·8-s + 0.366·9-s − 1.05·10-s − 0.320·11-s + 0.584·12-s − 0.137·13-s − 0.808·14-s − 1.73·15-s + 0.250·16-s + 1.02·17-s + 0.259·18-s + 0.335·19-s − 0.743·20-s − 1.33·21-s − 0.226·22-s + 1.84·23-s + 0.413·24-s + 1.20·25-s − 0.0972·26-s − 0.740·27-s − 0.571·28-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
| \(L(1)\) |
\(\approx\) |
\(2.928881694\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.928881694\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.02T + 3T^{2} \) |
| 5 | \( 1 + 3.32T + 5T^{2} \) |
| 7 | \( 1 + 3.02T + 7T^{2} \) |
| 11 | \( 1 + 1.06T + 11T^{2} \) |
| 13 | \( 1 + 0.496T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 8.84T + 23T^{2} \) |
| 29 | \( 1 + 2.59T + 29T^{2} \) |
| 31 | \( 1 - 3.24T + 31T^{2} \) |
| 37 | \( 1 - 7.01T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 - 8.57T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 + 7.13T + 59T^{2} \) |
| 61 | \( 1 + 4.05T + 61T^{2} \) |
| 67 | \( 1 - 1.05T + 67T^{2} \) |
| 71 | \( 1 - 0.547T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 4.58T + 79T^{2} \) |
| 83 | \( 1 - 4.13T + 83T^{2} \) |
| 89 | \( 1 - 10.7T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−8.238565692456200301850073410643, −7.61649200840067991757450066502, −7.27471614807578836512954273963, −6.28569186817010166745029852895, −5.36478452866655311335180326285, −4.36788392906693831214639765097, −3.65931937622632094000122830787, −3.08320027205267881125510064680, −2.61343034986922124244771757736, −0.820836073405446418968085483681,
0.820836073405446418968085483681, 2.61343034986922124244771757736, 3.08320027205267881125510064680, 3.65931937622632094000122830787, 4.36788392906693831214639765097, 5.36478452866655311335180326285, 6.28569186817010166745029852895, 7.27471614807578836512954273963, 7.61649200840067991757450066502, 8.238565692456200301850073410643
