| L(s) = 1 | − 3·7-s + 2·11-s + 5·13-s + 8·17-s − 19-s + 6·23-s + 2·29-s − 5·37-s − 10·41-s + 4·43-s + 4·47-s + 2·49-s + 2·53-s + 8·59-s + 7·61-s − 9·67-s − 2·71-s + 5·73-s − 6·77-s + 3·79-s + 6·83-s − 12·89-s − 15·91-s + 13·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 0.603·11-s + 1.38·13-s + 1.94·17-s − 0.229·19-s + 1.25·23-s + 0.371·29-s − 0.821·37-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 2/7·49-s + 0.274·53-s + 1.04·59-s + 0.896·61-s − 1.09·67-s − 0.237·71-s + 0.585·73-s − 0.683·77-s + 0.337·79-s + 0.658·83-s − 1.27·89-s − 1.57·91-s + 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.285065568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.285065568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 5 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−16.60458341729983, −15.94953554909279, −15.53786068655124, −14.77017870771841, −14.27278298536424, −13.56755972122390, −13.14945720197418, −12.49888691066553, −11.98042338973357, −11.38228191437826, −10.55156983427839, −10.16229286177554, −9.485150619298211, −8.846514245509144, −8.385454124357730, −7.486098914980845, −6.828879929776613, −6.295378316422849, −5.659160054585616, −4.984845185908468, −3.779754337451632, −3.522985831439891, −2.789214369630982, −1.489805552475195, −0.7667489445793734,
0.7667489445793734, 1.489805552475195, 2.789214369630982, 3.522985831439891, 3.779754337451632, 4.984845185908468, 5.659160054585616, 6.295378316422849, 6.828879929776613, 7.486098914980845, 8.385454124357730, 8.846514245509144, 9.485150619298211, 10.16229286177554, 10.55156983427839, 11.38228191437826, 11.98042338973357, 12.49888691066553, 13.14945720197418, 13.56755972122390, 14.27278298536424, 14.77017870771841, 15.53786068655124, 15.94953554909279, 16.60458341729983
