| L(s) = 1 | − 2.73·2-s + 0.112·3-s + 5.46·4-s + 0.682·5-s − 0.307·6-s − 0.658·7-s − 9.48·8-s − 2.98·9-s − 1.86·10-s + 1.29·11-s + 0.614·12-s + 1.76·13-s + 1.80·14-s + 0.0766·15-s + 14.9·16-s − 2.89·17-s + 8.16·18-s − 19-s + 3.73·20-s − 0.0740·21-s − 3.54·22-s + 1.86·23-s − 1.06·24-s − 4.53·25-s − 4.82·26-s − 0.673·27-s − 3.60·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.0648·3-s + 2.73·4-s + 0.305·5-s − 0.125·6-s − 0.248·7-s − 3.35·8-s − 0.995·9-s − 0.589·10-s + 0.391·11-s + 0.177·12-s + 0.489·13-s + 0.481·14-s + 0.0197·15-s + 3.74·16-s − 0.702·17-s + 1.92·18-s − 0.229·19-s + 0.834·20-s − 0.0161·21-s − 0.756·22-s + 0.389·23-s − 0.217·24-s − 0.906·25-s − 0.946·26-s − 0.129·27-s − 0.680·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 0.112T + 3T^{2} \) |
| 5 | \( 1 - 0.682T + 5T^{2} \) |
| 7 | \( 1 + 0.658T + 7T^{2} \) |
| 11 | \( 1 - 1.29T + 11T^{2} \) |
| 13 | \( 1 - 1.76T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 - 1.28T + 29T^{2} \) |
| 31 | \( 1 + 3.74T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 2.93T + 41T^{2} \) |
| 43 | \( 1 - 11.6T + 43T^{2} \) |
| 47 | \( 1 - 0.467T + 47T^{2} \) |
| 53 | \( 1 + 0.146T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 8.10T + 61T^{2} \) |
| 67 | \( 1 + 0.316T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 1.30T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 + 9.50T + 89T^{2} \) |
| 97 | \( 1 - 8.60T + 97T^{2} \) |
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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
−8.303158804891010174399570602521, −7.58954555384979577710709373643, −6.79089614690768152021103027307, −6.13981006436241453403727263464, −5.60469184387525316671381674778, −4.00774922883510942090714120177, −2.87784044488268137512972191885, −2.24598837138140454558943713796, −1.16643347205835832395182683269, 0,
1.16643347205835832395182683269, 2.24598837138140454558943713796, 2.87784044488268137512972191885, 4.00774922883510942090714120177, 5.60469184387525316671381674778, 6.13981006436241453403727263464, 6.79089614690768152021103027307, 7.58954555384979577710709373643, 8.303158804891010174399570602521
