| L(s) = 1 | + 2·2-s + 3.89·3-s + 4·4-s + 7.79·6-s + 8·8-s + 6.20·9-s − 21.6·11-s + 15.5·12-s + 16·16-s − 28.3·17-s + 12.4·18-s + 31.6·19-s − 43.3·22-s + 31.1·24-s − 10.9·27-s + 32·32-s − 84.5·33-s − 56.7·34-s + 24.8·36-s + 63.3·38-s + 81.7·41-s − 14·43-s − 86.7·44-s + 62.3·48-s + 49·49-s − 110.·51-s − 21.8·54-s + ⋯ |
| L(s) = 1 | + 2-s + 1.29·3-s + 4-s + 1.29·6-s + 8-s + 0.689·9-s − 1.97·11-s + 1.29·12-s + 16-s − 1.67·17-s + 0.689·18-s + 1.66·19-s − 1.97·22-s + 1.29·24-s − 0.404·27-s + 32-s − 2.56·33-s − 1.67·34-s + 0.689·36-s + 1.66·38-s + 1.99·41-s − 0.325·43-s − 1.97·44-s + 1.29·48-s + 0.999·49-s − 2.17·51-s − 0.404·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.730612124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.730612124\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.89T + 9T^{2} \) |
| 7 | \( 1 - 49T^{2} \) |
| 11 | \( 1 + 21.6T + 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 + 28.3T + 289T^{2} \) |
| 19 | \( 1 - 31.6T + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 81.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + 82T + 3.48e3T^{2} \) |
| 61 | \( 1 - 3.72e3T^{2} \) |
| 67 | \( 1 + 71.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 41.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 - 34.9T + 6.88e3T^{2} \) |
| 89 | \( 1 - 15.1T + 7.92e3T^{2} \) |
| 97 | \( 1 - 94T + 9.40e3T^{2} \) |
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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
−12.63774678137722186990813312318, −11.32396193677975787094525271085, −10.40571710707509908417704958965, −9.173860993119766088234785754812, −7.943804388025046803273624620282, −7.31175891169680837920549857042, −5.71495449471758822340502701116, −4.54731202775143293825911841871, −3.10906074882564947199058052685, −2.32514704188376471762340462203,
2.32514704188376471762340462203, 3.10906074882564947199058052685, 4.54731202775143293825911841871, 5.71495449471758822340502701116, 7.31175891169680837920549857042, 7.943804388025046803273624620282, 9.173860993119766088234785754812, 10.40571710707509908417704958965, 11.32396193677975787094525271085, 12.63774678137722186990813312318
