| L(s) = 1 | − 7-s − 3·11-s + 4·13-s − 2·19-s + 6·23-s − 6·29-s − 5·31-s − 2·37-s + 6·41-s − 10·43-s − 6·47-s − 6·49-s + 9·53-s + 12·59-s + 8·61-s + 14·67-s + 7·73-s + 3·77-s − 8·79-s + 3·83-s + 18·89-s − 4·91-s + 97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.904·11-s + 1.10·13-s − 0.458·19-s + 1.25·23-s − 1.11·29-s − 0.898·31-s − 0.328·37-s + 0.937·41-s − 1.52·43-s − 0.875·47-s − 6/7·49-s + 1.23·53-s + 1.56·59-s + 1.02·61-s + 1.71·67-s + 0.819·73-s + 0.341·77-s − 0.900·79-s + 0.329·83-s + 1.90·89-s − 0.419·91-s + 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-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
| \(L(1)\) |
\(\approx\) |
\(1.630312568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630312568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 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
−16.45527326002577, −16.00003072092604, −15.44195816034907, −14.80585497748430, −14.39224606543422, −13.36692092106529, −13.07049793418514, −12.85299985755080, −11.81664650767794, −11.18698490671543, −10.82854349964778, −10.11590754751317, −9.497934577905419, −8.789492474695365, −8.310249526893035, −7.589070337341718, −6.845939428182732, −6.355723043036496, −5.383547295419152, −5.134999287615746, −3.928611977554003, −3.487747926134016, −2.583916535492658, −1.725804113833233, −0.5894494978639125,
0.5894494978639125, 1.725804113833233, 2.583916535492658, 3.487747926134016, 3.928611977554003, 5.134999287615746, 5.383547295419152, 6.355723043036496, 6.845939428182732, 7.589070337341718, 8.310249526893035, 8.789492474695365, 9.497934577905419, 10.11590754751317, 10.82854349964778, 11.18698490671543, 11.81664650767794, 12.85299985755080, 13.07049793418514, 13.36692092106529, 14.39224606543422, 14.80585497748430, 15.44195816034907, 16.00003072092604, 16.45527326002577
