| L(s) = 1 | + 3-s − 2·9-s + 5·11-s + 5·17-s − 5·19-s + 6·23-s − 5·27-s − 4·29-s + 10·31-s + 5·33-s − 10·37-s − 5·41-s − 4·43-s + 8·47-s + 5·51-s − 10·53-s − 5·57-s − 10·61-s − 3·67-s + 6·69-s − 5·73-s − 10·79-s + 81-s − 83-s − 4·87-s + 9·89-s + 10·93-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 1.50·11-s + 1.21·17-s − 1.14·19-s + 1.25·23-s − 0.962·27-s − 0.742·29-s + 1.79·31-s + 0.870·33-s − 1.64·37-s − 0.780·41-s − 0.609·43-s + 1.16·47-s + 0.700·51-s − 1.37·53-s − 0.662·57-s − 1.28·61-s − 0.366·67-s + 0.722·69-s − 0.585·73-s − 1.12·79-s + 1/9·81-s − 0.109·83-s − 0.428·87-s + 0.953·89-s + 1.03·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.971914384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.971914384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.10321877776972, −13.60590670017227, −13.15518837976282, −12.41836593403126, −11.98413005948395, −11.69841724713455, −11.00647944518017, −10.51609446885372, −9.946235137611713, −9.358562643303932, −8.868878299038073, −8.594591504687521, −8.013628613535364, −7.371134222994186, −6.830402165689733, −6.260094767374611, −5.836347073024434, −5.071226457678495, −4.514759580138508, −3.830649174329180, −3.237264459726666, −2.924681545126334, −1.915432555407681, −1.445765172295934, −0.5454263907676235,
0.5454263907676235, 1.445765172295934, 1.915432555407681, 2.924681545126334, 3.237264459726666, 3.830649174329180, 4.514759580138508, 5.071226457678495, 5.836347073024434, 6.260094767374611, 6.830402165689733, 7.371134222994186, 8.013628613535364, 8.594591504687521, 8.868878299038073, 9.358562643303932, 9.946235137611713, 10.51609446885372, 11.00647944518017, 11.69841724713455, 11.98413005948395, 12.41836593403126, 13.15518837976282, 13.60590670017227, 14.10321877776972
