L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s + 6·13-s + 16-s − 2·17-s + 4·19-s + 4·22-s + 8·23-s + 6·26-s + 2·29-s + 32-s − 2·34-s + 10·37-s + 4·38-s − 6·41-s + 4·43-s + 4·44-s + 8·46-s + 6·52-s + 6·53-s + 2·58-s + 4·59-s − 6·61-s + 64-s − 4·67-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.852·22-s + 1.66·23-s + 1.17·26-s + 0.371·29-s + 0.176·32-s − 0.342·34-s + 1.64·37-s + 0.648·38-s − 0.937·41-s + 0.609·43-s + 0.603·44-s + 1.17·46-s + 0.832·52-s + 0.824·53-s + 0.262·58-s + 0.520·59-s − 0.768·61-s + 1/8·64-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.426703083\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.426703083\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 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 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.34104876834983, −15.06002144157067, −14.40098710326432, −13.81534539246061, −13.42814182545771, −12.97603395422598, −12.28618258685740, −11.64012784670004, −11.27160166982865, −10.85319717012877, −10.10057579273845, −9.280521304801093, −8.939755880179446, −8.304423901340515, −7.514315063302647, −6.856401000248984, −6.394466148818834, −5.852590283010547, −5.138561642835403, −4.412141519590022, −3.830318281173847, −3.257337598830280, −2.535732555657622, −1.376686704291773, −0.9927076417988643,
0.9927076417988643, 1.376686704291773, 2.535732555657622, 3.257337598830280, 3.830318281173847, 4.412141519590022, 5.138561642835403, 5.852590283010547, 6.394466148818834, 6.856401000248984, 7.514315063302647, 8.304423901340515, 8.939755880179446, 9.280521304801093, 10.10057579273845, 10.85319717012877, 11.27160166982865, 11.64012784670004, 12.28618258685740, 12.97603395422598, 13.42814182545771, 13.81534539246061, 14.40098710326432, 15.06002144157067, 15.34104876834983