| L(s) = 1 | − 2·3-s + 7-s + 9-s + 4·11-s − 6·17-s − 6·19-s − 2·21-s − 5·25-s + 4·27-s + 10·29-s − 4·31-s − 8·33-s + 2·37-s − 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s + 6·53-s + 12·57-s + 2·59-s + 63-s + 8·71-s − 6·73-s + 10·75-s + 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.45·17-s − 1.37·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.85·29-s − 0.718·31-s − 1.39·33-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s + 1.58·57-s + 0.260·59-s + 0.125·63-s + 0.949·71-s − 0.702·73-s + 1.15·75-s + 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6089543925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6089543925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−14.29077796608307, −13.88458959804547, −13.16596423350261, −12.81518591958445, −12.08282921262579, −11.71092900783473, −11.36054810919544, −10.96177931450693, −10.18464787987826, −10.02430724847635, −9.058218263641016, −8.549215904275185, −8.372503967396231, −7.351320503608160, −6.719252564654405, −6.365936773755588, −6.109363698113635, −5.133243556013725, −4.791959286252853, −4.194325371371835, −3.660371763974795, −2.668316613812702, −1.918131678642109, −1.314733814139281, −0.2944196793963340,
0.2944196793963340, 1.314733814139281, 1.918131678642109, 2.668316613812702, 3.660371763974795, 4.194325371371835, 4.791959286252853, 5.133243556013725, 6.109363698113635, 6.365936773755588, 6.719252564654405, 7.351320503608160, 8.372503967396231, 8.549215904275185, 9.058218263641016, 10.02430724847635, 10.18464787987826, 10.96177931450693, 11.36054810919544, 11.71092900783473, 12.08282921262579, 12.81518591958445, 13.16596423350261, 13.88458959804547, 14.29077796608307
