| L(s) = 1 | − 7-s + 6·11-s − 2·13-s + 4·17-s − 6·19-s − 2·29-s + 10·31-s − 4·37-s − 2·41-s + 4·43-s + 49-s − 6·53-s + 8·59-s + 2·61-s + 16·67-s + 10·71-s + 6·73-s − 6·77-s − 4·79-s + 8·83-s − 6·89-s + 2·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 1.80·11-s − 0.554·13-s + 0.970·17-s − 1.37·19-s − 0.371·29-s + 1.79·31-s − 0.657·37-s − 0.312·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.04·59-s + 0.256·61-s + 1.95·67-s + 1.18·71-s + 0.702·73-s − 0.683·77-s − 0.450·79-s + 0.878·83-s − 0.635·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.750660069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.750660069\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.99131830486704, −13.21625439724770, −12.60603765715514, −12.32770212997048, −11.88190485347592, −11.33845460585723, −10.87943663423267, −10.12383625915842, −9.812963660128926, −9.397355358180478, −8.736162075722692, −8.350785267047348, −7.801110709543934, −6.999786028503910, −6.687368655062269, −6.250217039593862, −5.680279311128745, −4.931698523299995, −4.422406496703263, −3.752042122665870, −3.467487947518730, −2.548331876412503, −2.003202408199730, −1.186674993566738, −0.5694863175137817,
0.5694863175137817, 1.186674993566738, 2.003202408199730, 2.548331876412503, 3.467487947518730, 3.752042122665870, 4.422406496703263, 4.931698523299995, 5.680279311128745, 6.250217039593862, 6.687368655062269, 6.999786028503910, 7.801110709543934, 8.350785267047348, 8.736162075722692, 9.397355358180478, 9.812963660128926, 10.12383625915842, 10.87943663423267, 11.33845460585723, 11.88190485347592, 12.32770212997048, 12.60603765715514, 13.21625439724770, 13.99131830486704
