| L(s) = 1 | + 2.20·7-s + 9·9-s + 20.3·11-s − 28.2·17-s + 19·19-s + 34.8·23-s + 67.0·43-s − 36.6·47-s − 44.1·49-s − 5.12·61-s + 19.8·63-s − 93.5·73-s + 44.9·77-s + 81·81-s − 139.·83-s + 183.·99-s + 102·101-s − 62.3·119-s + ⋯ |
| L(s) = 1 | + 0.315·7-s + 9-s + 1.85·11-s − 1.66·17-s + 19-s + 1.51·23-s + 1.55·43-s − 0.779·47-s − 0.900·49-s − 0.0839·61-s + 0.315·63-s − 1.28·73-s + 0.584·77-s + 81-s − 1.68·83-s + 1.85·99-s + 1.00·101-s − 0.524·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.749153408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.749153408\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 - 9T^{2} \) |
| 7 | \( 1 - 2.20T + 49T^{2} \) |
| 11 | \( 1 - 20.3T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 28.2T + 289T^{2} \) |
| 23 | \( 1 - 34.8T + 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 - 67.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 36.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 + 5.12T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−9.231894856620097472013543433220, −8.366732435380674312398780699481, −7.13839524552376924488964659626, −6.90911468194515699874588325867, −5.94508876560897468443575648188, −4.70624552092704869833303236310, −4.24028527506232148129055660998, −3.18940045971458959569856368548, −1.81738458010758069525836181031, −0.980906975324365483475661988521,
0.980906975324365483475661988521, 1.81738458010758069525836181031, 3.18940045971458959569856368548, 4.24028527506232148129055660998, 4.70624552092704869833303236310, 5.94508876560897468443575648188, 6.90911468194515699874588325867, 7.13839524552376924488964659626, 8.366732435380674312398780699481, 9.231894856620097472013543433220
