| L(s) = 1 | − 2·7-s + 6·11-s − 5·13-s − 3·17-s + 2·19-s + 6·23-s − 3·29-s − 4·31-s − 5·37-s + 6·41-s + 10·43-s − 3·49-s − 6·53-s + 12·59-s + 5·61-s − 2·67-s − 6·71-s + 73-s − 12·77-s − 10·79-s + 3·89-s + 10·91-s + 10·97-s − 6·101-s + 16·103-s + 12·107-s − 7·109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 1.80·11-s − 1.38·13-s − 0.727·17-s + 0.458·19-s + 1.25·23-s − 0.557·29-s − 0.718·31-s − 0.821·37-s + 0.937·41-s + 1.52·43-s − 3/7·49-s − 0.824·53-s + 1.56·59-s + 0.640·61-s − 0.244·67-s − 0.712·71-s + 0.117·73-s − 1.36·77-s − 1.12·79-s + 0.317·89-s + 1.04·91-s + 1.01·97-s − 0.597·101-s + 1.57·103-s + 1.16·107-s − 0.670·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722108016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722108016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.54573129342087076647481470812, −7.11469842903738619770774744094, −6.56731644873899858023492200478, −5.82737610324700040547902196988, −4.98718561034502658454296961526, −4.23085627135447810014209364493, −3.53394769247880554259259655842, −2.71824368801452204048848355470, −1.77311459833770670104944292484, −0.64771684037256430182658665863,
0.64771684037256430182658665863, 1.77311459833770670104944292484, 2.71824368801452204048848355470, 3.53394769247880554259259655842, 4.23085627135447810014209364493, 4.98718561034502658454296961526, 5.82737610324700040547902196988, 6.56731644873899858023492200478, 7.11469842903738619770774744094, 7.54573129342087076647481470812
