| L(s) = 1 | − 2.56·2-s − 1.95·3-s + 4.60·4-s + 1.92·5-s + 5.02·6-s + 3.09·7-s − 6.68·8-s + 0.830·9-s − 4.95·10-s + 6.15·11-s − 9.00·12-s + 6.36·13-s − 7.95·14-s − 3.77·15-s + 7.98·16-s − 2.90·17-s − 2.13·18-s − 0.236·19-s + 8.86·20-s − 6.05·21-s − 15.8·22-s + 7.62·23-s + 13.0·24-s − 1.28·25-s − 16.3·26-s + 4.24·27-s + 14.2·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 1.12·3-s + 2.30·4-s + 0.861·5-s + 2.05·6-s + 1.17·7-s − 2.36·8-s + 0.276·9-s − 1.56·10-s + 1.85·11-s − 2.60·12-s + 1.76·13-s − 2.12·14-s − 0.973·15-s + 1.99·16-s − 0.703·17-s − 0.502·18-s − 0.0541·19-s + 1.98·20-s − 1.32·21-s − 3.36·22-s + 1.58·23-s + 2.67·24-s − 0.257·25-s − 3.20·26-s + 0.817·27-s + 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.080567548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.080567548\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 53 | \( 1 + T \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 1.95T + 3T^{2} \) |
| 5 | \( 1 - 1.92T + 5T^{2} \) |
| 7 | \( 1 - 3.09T + 7T^{2} \) |
| 11 | \( 1 - 6.15T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 23 | \( 1 - 7.62T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 4.88T + 43T^{2} \) |
| 47 | \( 1 + 8.88T + 47T^{2} \) |
| 59 | \( 1 - 5.79T + 59T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 - 10.5T + 67T^{2} \) |
| 71 | \( 1 + 2.11T + 71T^{2} \) |
| 73 | \( 1 - 5.85T + 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 0.534T + 89T^{2} \) |
| 97 | \( 1 + 8.93T + 97T^{2} \) |
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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
−8.124442142185054214719295871395, −6.99168697755747364307250628622, −6.51652502470702213565737606045, −6.18271757875205219084749750288, −5.33706333216355705708148774906, −4.44678976111091238840442505561, −3.29249172436073802216912092239, −1.95229275436190924794563843966, −1.37627525882767553794956892716, −0.847671619074252199698520613157,
0.847671619074252199698520613157, 1.37627525882767553794956892716, 1.95229275436190924794563843966, 3.29249172436073802216912092239, 4.44678976111091238840442505561, 5.33706333216355705708148774906, 6.18271757875205219084749750288, 6.51652502470702213565737606045, 6.99168697755747364307250628622, 8.124442142185054214719295871395
