L(s) = 1 | − 2-s + 4-s + 5-s − 4·7-s − 8-s − 10-s + 13-s + 4·14-s + 16-s − 17-s − 4·19-s + 20-s + 25-s − 26-s − 4·28-s − 6·29-s − 4·31-s − 32-s + 34-s − 4·35-s + 2·37-s + 4·38-s − 40-s − 6·41-s − 4·43-s + 9·49-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s − 0.353·8-s − 0.316·10-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.917·19-s + 0.223·20-s + 1/5·25-s − 0.196·26-s − 0.755·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s − 0.676·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 0.937·41-s − 0.609·43-s + 9/7·49-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19890 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5810341480\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5810341480\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.95570149265086, −15.03390064834915, −14.90796215110039, −13.95247115103013, −13.31127453604250, −12.97046950922592, −12.48741190884352, −11.76725829377487, −11.08749101604532, −10.56489351845798, −10.01879169621713, −9.502595143816399, −9.074085963441593, −8.494920670752513, −7.792093097185218, −6.922824691263799, −6.693399563877043, −5.965559643658305, −5.525259924923441, −4.475655961031694, −3.639141219323810, −3.100506385531134, −2.265557956408865, −1.559437372860942, −0.3415309846634656,
0.3415309846634656, 1.559437372860942, 2.265557956408865, 3.100506385531134, 3.639141219323810, 4.475655961031694, 5.525259924923441, 5.965559643658305, 6.693399563877043, 6.922824691263799, 7.792093097185218, 8.494920670752513, 9.074085963441593, 9.502595143816399, 10.01879169621713, 10.56489351845798, 11.08749101604532, 11.76725829377487, 12.48741190884352, 12.97046950922592, 13.31127453604250, 13.95247115103013, 14.90796215110039, 15.03390064834915, 15.95570149265086