| L(s) = 1 | − 2-s − 2.00·3-s + 4-s + 5-s + 2.00·6-s − 0.566·7-s − 8-s + 1.03·9-s − 10-s − 3.22·11-s − 2.00·12-s + 1.91·13-s + 0.566·14-s − 2.00·15-s + 16-s + 6.98·17-s − 1.03·18-s + 7.08·19-s + 20-s + 1.13·21-s + 3.22·22-s + 5.48·23-s + 2.00·24-s + 25-s − 1.91·26-s + 3.94·27-s − 0.566·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.16·3-s + 0.5·4-s + 0.447·5-s + 0.820·6-s − 0.214·7-s − 0.353·8-s + 0.346·9-s − 0.316·10-s − 0.971·11-s − 0.580·12-s + 0.530·13-s + 0.151·14-s − 0.518·15-s + 0.250·16-s + 1.69·17-s − 0.245·18-s + 1.62·19-s + 0.223·20-s + 0.248·21-s + 0.687·22-s + 1.14·23-s + 0.410·24-s + 0.200·25-s − 0.375·26-s + 0.758·27-s − 0.107·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.023628511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.023628511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 + 2.00T + 3T^{2} \) |
| 7 | \( 1 + 0.566T + 7T^{2} \) |
| 11 | \( 1 + 3.22T + 11T^{2} \) |
| 13 | \( 1 - 1.91T + 13T^{2} \) |
| 17 | \( 1 - 6.98T + 17T^{2} \) |
| 19 | \( 1 - 7.08T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 3.17T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 + 2.88T + 41T^{2} \) |
| 43 | \( 1 - 2.55T + 43T^{2} \) |
| 47 | \( 1 - 0.387T + 47T^{2} \) |
| 53 | \( 1 + 2.94T + 53T^{2} \) |
| 59 | \( 1 - 0.646T + 59T^{2} \) |
| 61 | \( 1 - 15.3T + 61T^{2} \) |
| 67 | \( 1 - 1.36T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 6.99T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 + 0.856T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.136927258167897575546683177930, −7.25251668621723020876988483347, −6.74243593778525839731444079632, −5.82535527958914819649864540239, −5.41704495690840530437003597856, −4.87201613186477402079073643405, −3.32700678383498193520988314733, −2.84613868582776415862009114760, −1.38129411747053942440232834230, −0.70141497941942855384060173100,
0.70141497941942855384060173100, 1.38129411747053942440232834230, 2.84613868582776415862009114760, 3.32700678383498193520988314733, 4.87201613186477402079073643405, 5.41704495690840530437003597856, 5.82535527958914819649864540239, 6.74243593778525839731444079632, 7.25251668621723020876988483347, 8.136927258167897575546683177930
