| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 7-s + 8-s + 9-s + 12-s + 6·13-s + 14-s + 16-s − 2·17-s + 18-s + 6·19-s + 21-s + 23-s + 24-s − 5·25-s + 6·26-s + 27-s + 28-s + 6·29-s + 32-s − 2·34-s + 36-s + 6·38-s + 6·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 1.66·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 1.37·19-s + 0.218·21-s + 0.208·23-s + 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.176·32-s − 0.342·34-s + 1/6·36-s + 0.973·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116886 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.936827123\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.936827123\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.65461253042821, −13.26422845351515, −12.79035832137256, −12.11722949844660, −11.68581444872533, −11.28798277237816, −10.76794905364631, −10.24824372253445, −9.721122492140257, −9.077768988628977, −8.602820307011453, −8.211970589584744, −7.561752464316342, −7.178921281836621, −6.503817334429106, −5.975360894761091, −5.562436552721713, −4.827662358648993, −4.375985087030084, −3.659785912460190, −3.413565642675201, −2.673430335948367, −2.042771343652065, −1.341849223124612, −0.7843585716351931,
0.7843585716351931, 1.341849223124612, 2.042771343652065, 2.673430335948367, 3.413565642675201, 3.659785912460190, 4.375985087030084, 4.827662358648993, 5.562436552721713, 5.975360894761091, 6.503817334429106, 7.178921281836621, 7.561752464316342, 8.211970589584744, 8.602820307011453, 9.077768988628977, 9.721122492140257, 10.24824372253445, 10.76794905364631, 11.28798277237816, 11.68581444872533, 12.11722949844660, 12.79035832137256, 13.26422845351515, 13.65461253042821
