L(s) = 1 | + 1.39i·3-s + (−0.535 − 2.17i)5-s + (−2.13 + 2.13i)7-s + 1.05·9-s + (−2.17 + 2.17i)11-s − 1.54·13-s + (3.02 − 0.745i)15-s + (−3.86 + 3.86i)17-s + (0.0136 − 0.0136i)19-s + (−2.97 − 2.97i)21-s + (−3.15 − 3.15i)23-s + (−4.42 + 2.32i)25-s + 5.65i·27-s + (3.33 + 3.33i)29-s + 8.92i·31-s + ⋯ |
L(s) = 1 | + 0.804i·3-s + (−0.239 − 0.970i)5-s + (−0.806 + 0.806i)7-s + 0.353·9-s + (−0.654 + 0.654i)11-s − 0.428·13-s + (0.780 − 0.192i)15-s + (−0.937 + 0.937i)17-s + (0.00313 − 0.00313i)19-s + (−0.648 − 0.648i)21-s + (−0.657 − 0.657i)23-s + (−0.885 + 0.464i)25-s + 1.08i·27-s + (0.619 + 0.619i)29-s + 1.60i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.156095 + 0.629697i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.156095 + 0.629697i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.535 + 2.17i)T \) |
good | 3 | \( 1 - 1.39iT - 3T^{2} \) |
| 7 | \( 1 + (2.13 - 2.13i)T - 7iT^{2} \) |
| 11 | \( 1 + (2.17 - 2.17i)T - 11iT^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 + (3.86 - 3.86i)T - 17iT^{2} \) |
| 19 | \( 1 + (-0.0136 + 0.0136i)T - 19iT^{2} \) |
| 23 | \( 1 + (3.15 + 3.15i)T + 23iT^{2} \) |
| 29 | \( 1 + (-3.33 - 3.33i)T + 29iT^{2} \) |
| 31 | \( 1 - 8.92iT - 31T^{2} \) |
| 37 | \( 1 + 7.24T + 37T^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + (3.34 + 3.34i)T + 47iT^{2} \) |
| 53 | \( 1 - 7.30iT - 53T^{2} \) |
| 59 | \( 1 + (3.52 + 3.52i)T + 59iT^{2} \) |
| 61 | \( 1 + (1.41 - 1.41i)T - 61iT^{2} \) |
| 67 | \( 1 + 0.748T + 67T^{2} \) |
| 71 | \( 1 + 0.269T + 71T^{2} \) |
| 73 | \( 1 + (0.811 - 0.811i)T - 73iT^{2} \) |
| 79 | \( 1 - 2.80T + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + (-6.33 + 6.33i)T - 97iT^{2} \) |
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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
−10.52338966119361725533841573374, −10.20511126921125809035031448259, −9.013762245567135512844470275805, −8.775153319423575261403395319668, −7.47345197305982606135247672166, −6.38517611691212366649653653626, −5.19665118830480250477605456228, −4.58241746940784323039349272261, −3.50275217434407591653345803070, −2.01737750290950670755329915064,
0.33059600635445920082049906134, 2.28665586216818033584261691685, 3.32996360132752220141600802613, 4.47504457626739637308014254883, 6.05067124378634780010532257598, 6.75318400863341211098721725862, 7.44192485958160270873803721318, 8.092900078847511994115692283131, 9.628631039269295150041267925099, 10.14076781232573608068781791786