| L(s) = 1 | − 5-s + 2·11-s − 2·13-s − 2·17-s − 23-s + 25-s + 29-s + 2·31-s + 4·37-s + 5·41-s + 11·43-s + 8·47-s + 8·53-s − 2·55-s − 4·59-s + 5·61-s + 2·65-s − 67-s + 12·71-s − 6·73-s + 4·79-s + 13·83-s + 2·85-s − 3·89-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.603·11-s − 0.554·13-s − 0.485·17-s − 0.208·23-s + 1/5·25-s + 0.185·29-s + 0.359·31-s + 0.657·37-s + 0.780·41-s + 1.67·43-s + 1.16·47-s + 1.09·53-s − 0.269·55-s − 0.520·59-s + 0.640·61-s + 0.248·65-s − 0.122·67-s + 1.42·71-s − 0.702·73-s + 0.450·79-s + 1.42·83-s + 0.216·85-s − 0.317·89-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.247726603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.247726603\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 13 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−14.23750164406171, −13.62929405760025, −13.16045911738981, −12.48267159578905, −12.15233630869415, −11.73562930270986, −10.98513178883845, −10.82121992455825, −10.04548885732440, −9.517653745019028, −9.070421063812783, −8.538831613988592, −7.959251253799874, −7.322516462130442, −7.085261389374178, −6.186267091889372, −5.956945649483204, −5.070307518676580, −4.545353624149431, −4.002438476437809, −3.518545476635661, −2.524424130310923, −2.286562947177916, −1.146187527216308, −0.5595006913753286,
0.5595006913753286, 1.146187527216308, 2.286562947177916, 2.524424130310923, 3.518545476635661, 4.002438476437809, 4.545353624149431, 5.070307518676580, 5.956945649483204, 6.186267091889372, 7.085261389374178, 7.322516462130442, 7.959251253799874, 8.538831613988592, 9.070421063812783, 9.517653745019028, 10.04548885732440, 10.82121992455825, 10.98513178883845, 11.73562930270986, 12.15233630869415, 12.48267159578905, 13.16045911738981, 13.62929405760025, 14.23750164406171
