L(s) = 1 | + (−0.363 + 1.96i)2-s + (−3.73 − 1.42i)4-s − 0.0632·5-s + 2.71i·7-s + (4.16 − 6.82i)8-s + (0.0229 − 0.124i)10-s + 1.99i·11-s + 9.23·13-s + (−5.34 − 0.987i)14-s + (11.9 + 10.6i)16-s − 29.7·17-s − 4.35i·19-s + (0.236 + 0.0904i)20-s + (−3.91 − 0.723i)22-s + 30.0i·23-s + ⋯ |
L(s) = 1 | + (−0.181 + 0.983i)2-s + (−0.934 − 0.357i)4-s − 0.0126·5-s + 0.388i·7-s + (0.520 − 0.853i)8-s + (0.00229 − 0.0124i)10-s + 0.180i·11-s + 0.710·13-s + (−0.381 − 0.0705i)14-s + (0.744 + 0.667i)16-s − 1.75·17-s − 0.229i·19-s + (0.0118 + 0.00452i)20-s + (−0.177 − 0.0328i)22-s + 1.30i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.357 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.05294248016\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05294248016\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.363 - 1.96i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + 4.35iT \) |
good | 5 | \( 1 + 0.0632T + 25T^{2} \) |
| 7 | \( 1 - 2.71iT - 49T^{2} \) |
| 11 | \( 1 - 1.99iT - 121T^{2} \) |
| 13 | \( 1 - 9.23T + 169T^{2} \) |
| 17 | \( 1 + 29.7T + 289T^{2} \) |
| 23 | \( 1 - 30.0iT - 529T^{2} \) |
| 29 | \( 1 - 1.04T + 841T^{2} \) |
| 31 | \( 1 + 24.8iT - 961T^{2} \) |
| 37 | \( 1 + 22.4T + 1.36e3T^{2} \) |
| 41 | \( 1 + 32.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 48.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 13.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 87.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 74.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 31.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 115. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 66.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 74.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 45.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 4.33iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 85.9T + 9.40e3T^{2} \) |
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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
−10.79548015008918395618739364804, −9.684990774592570033265694264716, −9.037011534225902874221231430992, −8.275587021875063856656505898909, −7.34181265182750042912766648275, −6.45902188904676779449033327924, −5.67514352515298334733637899129, −4.64896527758861260660677489517, −3.63416120103203350630395377151, −1.84414021339971282367308492291,
0.02015723149158042947614319456, 1.53995981577732158757522937662, 2.77722799663516420933693405599, 3.96597417524861786480511310452, 4.69513388101252972483497206879, 6.04283942195111040790729841443, 7.10937600031289069546535680533, 8.357296317020132062914765338716, 8.795020342918059480469414927944, 9.854805649710397706514010350184