| L(s) = 1 | + 7-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s + 10·29-s − 6·37-s + 6·41-s + 4·43-s − 8·47-s + 49-s + 6·53-s + 4·59-s − 10·61-s − 4·67-s + 16·71-s + 14·73-s + 4·77-s + 8·79-s − 4·83-s − 10·89-s + 2·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s + 1.85·29-s − 0.986·37-s + 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.28·61-s − 0.488·67-s + 1.89·71-s + 1.63·73-s + 0.455·77-s + 0.900·79-s − 0.439·83-s − 1.05·89-s + 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.710055325\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.710055325\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−16.31532934404926, −15.73897408563640, −15.15405769143067, −14.54787152386021, −14.05652947688921, −13.68029314361573, −12.80173562560280, −12.21518671235812, −11.89529020506881, −10.99460024775745, −10.75709318636741, −9.884372932538912, −9.367158505952335, −8.537363844405422, −8.339703878622892, −7.452416046438638, −6.622317540065626, −6.345103748771763, −5.485730616905278, −4.700512870880155, −4.069986381915312, −3.426148244382005, −2.476442912432571, −1.563432284596902, −0.7980328770170860,
0.7980328770170860, 1.563432284596902, 2.476442912432571, 3.426148244382005, 4.069986381915312, 4.700512870880155, 5.485730616905278, 6.345103748771763, 6.622317540065626, 7.452416046438638, 8.339703878622892, 8.537363844405422, 9.367158505952335, 9.884372932538912, 10.75709318636741, 10.99460024775745, 11.89529020506881, 12.21518671235812, 12.80173562560280, 13.68029314361573, 14.05652947688921, 14.54787152386021, 15.15405769143067, 15.73897408563640, 16.31532934404926
