| L(s) = 1 | + i·2-s − 2.82·3-s − 4-s + 1.41i·5-s − 2.82i·6-s − i·8-s + 5.00·9-s − 1.41·10-s + 2.82·12-s + (0.707 + 3.53i)13-s − 4.00i·15-s + 16-s − 1.41·17-s + 5.00i·18-s + 2.82i·19-s − 1.41i·20-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 1.63·3-s − 0.5·4-s + 0.632i·5-s − 1.15i·6-s − 0.353i·8-s + 1.66·9-s − 0.447·10-s + 0.816·12-s + (0.196 + 0.980i)13-s − 1.03i·15-s + 0.250·16-s − 0.342·17-s + 1.17i·18-s + 0.648i·19-s − 0.316i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.980 + 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5409145041\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5409145041\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (-0.707 - 3.53i)T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 1.41T + 17T^{2} \) |
| 19 | \( 1 - 2.82iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 9.89iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + 11.3iT - 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 2.82iT - 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 8iT - 71T^{2} \) |
| 73 | \( 1 + 12.7iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 2.82iT - 83T^{2} \) |
| 89 | \( 1 - 12.7iT - 89T^{2} \) |
| 97 | \( 1 + 4.24iT - 97T^{2} \) |
| 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
−10.23623678925172220789277546047, −9.377071705500032244570525865814, −8.379555242255389438452620436069, −7.23878262718331290824837497453, −6.57418160590708023108157359047, −6.25484721322355110924859261719, −5.06807163456452055141954137976, −4.61554347102087897554006756145, −3.26326268199755235476918631218, −1.36008864948723964142206053887,
0.34025819112147978708959259724, 1.29260799294295202844084477994, 2.93769580015295571114976294462, 4.28573809724012797728313963979, 5.07565590706356663347037242251, 5.56559789247720039262800349742, 6.60883728236905342636592464929, 7.49374762611980352894078129789, 8.713264299919191245312360289561, 9.350022318897502030482759183425
