L(s) = 1 | + 2-s + 0.988·3-s + 4-s + 5-s + 0.988·6-s − 3.47·7-s + 8-s − 2.02·9-s + 10-s + 6.04·11-s + 0.988·12-s − 3.47·14-s + 0.988·15-s + 16-s + 2.05·17-s − 2.02·18-s + 3.11·19-s + 20-s − 3.43·21-s + 6.04·22-s + 7.97·23-s + 0.988·24-s + 25-s − 4.96·27-s − 3.47·28-s − 6.73·29-s + 0.988·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.570·3-s + 0.5·4-s + 0.447·5-s + 0.403·6-s − 1.31·7-s + 0.353·8-s − 0.674·9-s + 0.316·10-s + 1.82·11-s + 0.285·12-s − 0.928·14-s + 0.255·15-s + 0.250·16-s + 0.498·17-s − 0.476·18-s + 0.715·19-s + 0.223·20-s − 0.749·21-s + 1.28·22-s + 1.66·23-s + 0.201·24-s + 0.200·25-s − 0.955·27-s − 0.656·28-s − 1.24·29-s + 0.180·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.370089046\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.370089046\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 0.988T + 3T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 11 | \( 1 - 6.04T + 11T^{2} \) |
| 17 | \( 1 - 2.05T + 17T^{2} \) |
| 19 | \( 1 - 3.11T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 - 8.96T + 31T^{2} \) |
| 37 | \( 1 - 2.69T + 37T^{2} \) |
| 41 | \( 1 - 3.23T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 + 11.1T + 53T^{2} \) |
| 59 | \( 1 + 6.40T + 59T^{2} \) |
| 61 | \( 1 - 13.0T + 61T^{2} \) |
| 67 | \( 1 + 8.49T + 67T^{2} \) |
| 71 | \( 1 - 0.121T + 71T^{2} \) |
| 73 | \( 1 + 0.138T + 73T^{2} \) |
| 79 | \( 1 + 6.71T + 79T^{2} \) |
| 83 | \( 1 + 3.03T + 83T^{2} \) |
| 89 | \( 1 - 3.38T + 89T^{2} \) |
| 97 | \( 1 + 5.82T + 97T^{2} \) |
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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
−9.352365714764957490930383976528, −8.807099246619844815609646546331, −7.58944705276073914779773563352, −6.71113631294560119140825406138, −6.17800629522632903888918770816, −5.36785402469041419754350100123, −4.11410225039249633519070382506, −3.29426076875458060590862528826, −2.72538522660946715358389260486, −1.22317563075328088421290407245,
1.22317563075328088421290407245, 2.72538522660946715358389260486, 3.29426076875458060590862528826, 4.11410225039249633519070382506, 5.36785402469041419754350100123, 6.17800629522632903888918770816, 6.71113631294560119140825406138, 7.58944705276073914779773563352, 8.807099246619844815609646546331, 9.352365714764957490930383976528