| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 2·11-s + 12-s + 13-s − 16-s − 6·17-s + 18-s − 4·19-s + 2·22-s + 4·23-s + 3·24-s + 26-s − 27-s − 6·29-s + 4·31-s + 5·32-s − 2·33-s − 6·34-s − 36-s − 10·37-s − 4·38-s − 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.603·11-s + 0.288·12-s + 0.277·13-s − 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.426·22-s + 0.834·23-s + 0.612·24-s + 0.196·26-s − 0.192·27-s − 1.11·29-s + 0.718·31-s + 0.883·32-s − 0.348·33-s − 1.02·34-s − 1/6·36-s − 1.64·37-s − 0.648·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.133774472\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133774472\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.44979650067160, −14.03367494769003, −13.46341294815201, −13.05608371529622, −12.62428689201857, −12.07368151259079, −11.62303473754941, −10.86702334833264, −10.74887210109305, −9.891728799630050, −9.191801576480179, −8.887376888611186, −8.473703053513545, −7.536917437029278, −6.877242892448590, −6.495368357323603, −5.781910978951993, −5.482570760944619, −4.504488627681876, −4.392420517133659, −3.762017732673505, −2.978409984966103, −2.214785645731448, −1.340633502323594, −0.3620130810338730,
0.3620130810338730, 1.340633502323594, 2.214785645731448, 2.978409984966103, 3.762017732673505, 4.392420517133659, 4.504488627681876, 5.482570760944619, 5.781910978951993, 6.495368357323603, 6.877242892448590, 7.536917437029278, 8.473703053513545, 8.887376888611186, 9.191801576480179, 9.891728799630050, 10.74887210109305, 10.86702334833264, 11.62303473754941, 12.07368151259079, 12.62428689201857, 13.05608371529622, 13.46341294815201, 14.03367494769003, 14.44979650067160
