| L(s) = 1 | + 2·7-s + 4·11-s + 13-s + 8·17-s + 6·19-s − 6·23-s + 4·29-s + 2·37-s + 2·41-s − 4·43-s − 3·49-s − 10·53-s + 4·59-s − 10·61-s + 12·67-s − 8·71-s + 8·73-s + 8·77-s − 8·79-s − 12·83-s + 14·89-s + 2·91-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 0.277·13-s + 1.94·17-s + 1.37·19-s − 1.25·23-s + 0.742·29-s + 0.328·37-s + 0.312·41-s − 0.609·43-s − 3/7·49-s − 1.37·53-s + 0.520·59-s − 1.28·61-s + 1.46·67-s − 0.949·71-s + 0.936·73-s + 0.911·77-s − 0.900·79-s − 1.31·83-s + 1.48·89-s + 0.209·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.814397846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.814397846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.44410015859292, −14.10382747394510, −13.91095028434095, −13.02819656188080, −12.36434839820494, −11.96046643923441, −11.60862413080823, −11.13830300197299, −10.34874278291171, −9.765876624046945, −9.597674716993578, −8.732574674214248, −8.239453528792638, −7.664654454646305, −7.360959053779086, −6.410232911643341, −6.043697964969649, −5.373250677828592, −4.818735079381951, −4.147286172182267, −3.438391635772257, −3.075684303389835, −1.923198472222481, −1.357694032103301, −0.7747762876970657,
0.7747762876970657, 1.357694032103301, 1.923198472222481, 3.075684303389835, 3.438391635772257, 4.147286172182267, 4.818735079381951, 5.373250677828592, 6.043697964969649, 6.410232911643341, 7.360959053779086, 7.664654454646305, 8.239453528792638, 8.732574674214248, 9.597674716993578, 9.765876624046945, 10.34874278291171, 11.13830300197299, 11.60862413080823, 11.96046643923441, 12.36434839820494, 13.02819656188080, 13.91095028434095, 14.10382747394510, 14.44410015859292
