L(s) = 1 | + (−773. + 671. i)2-s + 4.45e4i·3-s + (1.47e5 − 1.03e6i)4-s + 1.43e7·5-s + (−2.99e7 − 3.44e7i)6-s − 2.43e8i·7-s + (5.82e8 + 9.02e8i)8-s + 1.50e9·9-s + (−1.11e10 + 9.64e9i)10-s + 3.70e10i·11-s + (4.62e10 + 6.58e9i)12-s + 2.55e9·13-s + (1.63e11 + 1.88e11i)14-s + 6.40e11i·15-s + (−1.05e12 − 3.06e11i)16-s + 1.09e12·17-s + ⋯ |
L(s) = 1 | + (−0.755 + 0.655i)2-s + 0.754i·3-s + (0.140 − 0.990i)4-s + 1.47·5-s + (−0.494 − 0.570i)6-s − 0.863i·7-s + (0.542 + 0.840i)8-s + 0.430·9-s + (−1.11 + 0.964i)10-s + 1.42i·11-s + (0.747 + 0.106i)12-s + 0.0185·13-s + (0.565 + 0.651i)14-s + 1.11i·15-s + (−0.960 − 0.278i)16-s + 0.543·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{21}{2})\) |
\(\approx\) |
\(1.20202 + 1.04310i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.20202 + 1.04310i\) |
\(L(11)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (773. - 671. i)T \) |
good | 3 | \( 1 - 4.45e4iT - 3.48e9T^{2} \) |
| 5 | \( 1 - 1.43e7T + 9.53e13T^{2} \) |
| 7 | \( 1 + 2.43e8iT - 7.97e16T^{2} \) |
| 11 | \( 1 - 3.70e10iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 2.55e9T + 1.90e22T^{2} \) |
| 17 | \( 1 - 1.09e12T + 4.06e24T^{2} \) |
| 19 | \( 1 + 5.54e12iT - 3.75e25T^{2} \) |
| 23 | \( 1 - 6.68e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 - 5.41e14T + 1.76e29T^{2} \) |
| 31 | \( 1 - 2.99e14iT - 6.71e29T^{2} \) |
| 37 | \( 1 - 2.28e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 2.45e16T + 1.80e32T^{2} \) |
| 43 | \( 1 - 3.95e15iT - 4.67e32T^{2} \) |
| 47 | \( 1 + 4.94e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 - 1.12e17T + 3.05e34T^{2} \) |
| 59 | \( 1 + 5.57e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 + 7.00e17T + 5.08e35T^{2} \) |
| 67 | \( 1 + 2.27e18iT - 3.32e36T^{2} \) |
| 71 | \( 1 - 1.75e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 1.55e18T + 1.84e37T^{2} \) |
| 79 | \( 1 - 1.92e18iT - 8.96e37T^{2} \) |
| 83 | \( 1 + 1.23e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 + 1.93e19T + 9.72e38T^{2} \) |
| 97 | \( 1 + 6.55e18T + 5.43e39T^{2} \) |
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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
−20.12946183876646400799028617228, −17.93352740625767104025593981198, −16.99143650777994344452513102064, −15.31759557879542925033635855851, −13.68769899077199866418882965771, −10.26698139581514799376273672903, −9.579658435742864381368455278090, −6.99877015531722100227205922241, −4.98285899966701563000550630050, −1.54756488410906826410757658970,
1.14912975768301751997692177784, 2.53200668519542662450449023702, 6.20299174905374494876183689890, 8.526572883935977668435616682796, 10.19321535552977804651335880549, 12.29962180085372198822998151699, 13.68095767243831730275010239599, 16.61146728308595036616083973135, 18.20925600179283103352834532712, 18.87363016381420794039266243957