| L(s) = 1 | + 2-s + 2.97·3-s + 4-s − 3.65·5-s + 2.97·6-s − 3.15·7-s + 8-s + 5.85·9-s − 3.65·10-s − 2.20·11-s + 2.97·12-s + 6.40·13-s − 3.15·14-s − 10.8·15-s + 16-s − 7.51·17-s + 5.85·18-s − 4.30·19-s − 3.65·20-s − 9.38·21-s − 2.20·22-s − 2.63·23-s + 2.97·24-s + 8.38·25-s + 6.40·26-s + 8.48·27-s − 3.15·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.71·3-s + 0.5·4-s − 1.63·5-s + 1.21·6-s − 1.19·7-s + 0.353·8-s + 1.95·9-s − 1.15·10-s − 0.665·11-s + 0.858·12-s + 1.77·13-s − 0.842·14-s − 2.81·15-s + 0.250·16-s − 1.82·17-s + 1.37·18-s − 0.988·19-s − 0.817·20-s − 2.04·21-s − 0.470·22-s − 0.549·23-s + 0.607·24-s + 1.67·25-s + 1.25·26-s + 1.63·27-s − 0.595·28-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 - 2.97T + 3T^{2} \) |
| 5 | \( 1 + 3.65T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 + 2.20T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 7.51T + 17T^{2} \) |
| 19 | \( 1 + 4.30T + 19T^{2} \) |
| 23 | \( 1 + 2.63T + 23T^{2} \) |
| 29 | \( 1 + 4.67T + 29T^{2} \) |
| 31 | \( 1 + 4.24T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 + 10.8T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 - 6.45T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.15T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 0.761T + 71T^{2} \) |
| 73 | \( 1 - 1.30T + 73T^{2} \) |
| 79 | \( 1 - 0.548T + 79T^{2} \) |
| 83 | \( 1 + 5.63T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 3.73T + 97T^{2} \) |
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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
−8.114138341664056959585155986719, −7.40454106041497677498868983293, −6.79176233087680623057356521668, −6.03071923121255381518621093449, −4.57297175747538140558012672755, −3.81476457312291816797160517662, −3.66272164475576099478040362306, −2.82809186787931375398679779126, −1.90143919776736998538238911226, 0,
1.90143919776736998538238911226, 2.82809186787931375398679779126, 3.66272164475576099478040362306, 3.81476457312291816797160517662, 4.57297175747538140558012672755, 6.03071923121255381518621093449, 6.79176233087680623057356521668, 7.40454106041497677498868983293, 8.114138341664056959585155986719
