L(s) = 1 | + 1.41i·2-s − 5.41i·3-s − 2.00·4-s + 7.66·6-s − 8.24·7-s − 2.82i·8-s − 20.3·9-s − 15.8i·11-s + 10.8i·12-s − 14.3i·13-s − 11.6i·14-s + 4.00·16-s + 10.1·17-s − 28.8i·18-s − 36.5i·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.80i·3-s − 0.500·4-s + 1.27·6-s − 1.17·7-s − 0.353i·8-s − 2.26·9-s − 1.43i·11-s + 0.903i·12-s − 1.10i·13-s − 0.832i·14-s + 0.250·16-s + 0.598·17-s − 1.60i·18-s − 1.92i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5734869102\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5734869102\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 1.41iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 + (-5.84 - 22.2i)T \) |
good | 3 | \( 1 + 5.41iT - 9T^{2} \) |
| 7 | \( 1 + 8.24T + 49T^{2} \) |
| 11 | \( 1 + 15.8iT - 121T^{2} \) |
| 13 | \( 1 + 14.3iT - 169T^{2} \) |
| 17 | \( 1 - 10.1T + 289T^{2} \) |
| 19 | \( 1 + 36.5iT - 361T^{2} \) |
| 29 | \( 1 + 6.46T + 841T^{2} \) |
| 31 | \( 1 + 42.8T + 961T^{2} \) |
| 37 | \( 1 - 63.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.00T + 1.84e3T^{2} \) |
| 47 | \( 1 - 32.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 6.65T + 3.48e3T^{2} \) |
| 61 | \( 1 - 55.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 4.45T + 4.48e3T^{2} \) |
| 71 | \( 1 - 118.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 82.2iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 133. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 67.5T + 6.88e3T^{2} \) |
| 89 | \( 1 - 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 98.6T + 9.40e3T^{2} \) |
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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
−8.846717595106741328053315140710, −7.983470980822108047681728967180, −7.37652424494484555043846912388, −6.62149897073662169515831037080, −5.94104539244640185777737044875, −5.35812463680895205849546020618, −3.37469298805678506035394404536, −2.77022342487484879397291434587, −0.975089854818079972994523632449, −0.20804400515207799442388419970,
2.04654846066844308254901950666, 3.31031582030877198039608464512, 3.93868110369208437828771415989, 4.64374455271030809823886187777, 5.61976873697064575707121030922, 6.61111380028720674267159715401, 7.941870973435789671788052567333, 9.052128913444928234495734580269, 9.652315819364529933419949087324, 9.941859226492720282585192571576