| L(s) = 1 | + 1.14·2-s − 1.66·3-s − 0.689·4-s − 2.63·5-s − 1.90·6-s − 1.31·7-s − 3.07·8-s − 0.222·9-s − 3.01·10-s + 3.37·11-s + 1.14·12-s − 0.479·13-s − 1.50·14-s + 4.38·15-s − 2.14·16-s + 17-s − 0.254·18-s + 1.94·19-s + 1.81·20-s + 2.19·21-s + 3.85·22-s + 1.96·23-s + 5.13·24-s + 1.93·25-s − 0.548·26-s + 5.37·27-s + 0.909·28-s + ⋯ |
| L(s) = 1 | + 0.809·2-s − 0.962·3-s − 0.344·4-s − 1.17·5-s − 0.778·6-s − 0.498·7-s − 1.08·8-s − 0.0741·9-s − 0.953·10-s + 1.01·11-s + 0.331·12-s − 0.132·13-s − 0.403·14-s + 1.13·15-s − 0.536·16-s + 0.242·17-s − 0.0600·18-s + 0.447·19-s + 0.405·20-s + 0.479·21-s + 0.822·22-s + 0.409·23-s + 1.04·24-s + 0.386·25-s − 0.107·26-s + 1.03·27-s + 0.171·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8302152613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8302152613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 2 | \( 1 - 1.14T + 2T^{2} \) |
| 3 | \( 1 + 1.66T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 1.31T + 7T^{2} \) |
| 11 | \( 1 - 3.37T + 11T^{2} \) |
| 13 | \( 1 + 0.479T + 13T^{2} \) |
| 19 | \( 1 - 1.94T + 19T^{2} \) |
| 23 | \( 1 - 1.96T + 23T^{2} \) |
| 29 | \( 1 + 4.00T + 29T^{2} \) |
| 31 | \( 1 - 6.62T + 31T^{2} \) |
| 37 | \( 1 - 0.120T + 37T^{2} \) |
| 41 | \( 1 - 1.55T + 41T^{2} \) |
| 43 | \( 1 + 0.981T + 43T^{2} \) |
| 47 | \( 1 - 2.25T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 61 | \( 1 - 5.98T + 61T^{2} \) |
| 67 | \( 1 + 1.47T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 + 1.92T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 4.08T + 83T^{2} \) |
| 89 | \( 1 + 6.06T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−10.03855242975575529555271528401, −9.133078869283662625620791319287, −8.332828337145796588980714508446, −7.20056134509166357445711971310, −6.34546242769630310395147439832, −5.58774544368496284798692844691, −4.65888469963146902113828374231, −3.88848891115287131194013347141, −3.06032838627844108370074183752, −0.64829671702544017661164546831,
0.64829671702544017661164546831, 3.06032838627844108370074183752, 3.88848891115287131194013347141, 4.65888469963146902113828374231, 5.58774544368496284798692844691, 6.34546242769630310395147439832, 7.20056134509166357445711971310, 8.332828337145796588980714508446, 9.133078869283662625620791319287, 10.03855242975575529555271528401
