| L(s) = 1 | − 5·7-s − 5·11-s + 13-s − 3·17-s − 4·19-s − 5·23-s − 4·29-s − 7·37-s − 11·41-s + 12·43-s − 6·47-s + 18·49-s − 53-s − 12·59-s + 7·61-s − 4·67-s − 7·71-s + 14·73-s + 25·77-s + 5·79-s − 2·83-s + 3·89-s − 5·91-s + 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.88·7-s − 1.50·11-s + 0.277·13-s − 0.727·17-s − 0.917·19-s − 1.04·23-s − 0.742·29-s − 1.15·37-s − 1.71·41-s + 1.82·43-s − 0.875·47-s + 18/7·49-s − 0.137·53-s − 1.56·59-s + 0.896·61-s − 0.488·67-s − 0.830·71-s + 1.63·73-s + 2.84·77-s + 0.562·79-s − 0.219·83-s + 0.317·89-s − 0.524·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.24670777021240, −12.95257808235974, −12.37882568251914, −12.31541393573097, −11.38827482365339, −10.91282620231018, −10.38855066627726, −10.17204789115895, −9.642078898206039, −9.071052695206875, −8.713023967415252, −8.081733126202525, −7.609035798767324, −7.029412905650993, −6.477480835436962, −6.221654357219600, −5.593776214209412, −5.138742415159193, −4.401350371937550, −3.798846471355767, −3.394891778331966, −2.740944156482537, −2.305363667827032, −1.657424185199093, −0.4158300861552570, 0,
0.4158300861552570, 1.657424185199093, 2.305363667827032, 2.740944156482537, 3.394891778331966, 3.798846471355767, 4.401350371937550, 5.138742415159193, 5.593776214209412, 6.221654357219600, 6.477480835436962, 7.029412905650993, 7.609035798767324, 8.081733126202525, 8.713023967415252, 9.071052695206875, 9.642078898206039, 10.17204789115895, 10.38855066627726, 10.91282620231018, 11.38827482365339, 12.31541393573097, 12.37882568251914, 12.95257808235974, 13.24670777021240
