L(s) = 1 | + (−0.5 − 0.866i)3-s + (1.32 + 2.29i)7-s + (−0.499 + 0.866i)9-s + (0.177 + 0.306i)11-s − 13-s + (1.82 + 3.15i)17-s + (−0.322 + 0.559i)19-s + (1.32 − 2.29i)21-s + (−1 + 1.73i)23-s + 0.999·27-s − 3.29·29-s + (−2 − 3.46i)31-s + (0.177 − 0.306i)33-s + (−1.14 + 1.98i)37-s + (0.5 + 0.866i)39-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.499 + 0.866i)7-s + (−0.166 + 0.288i)9-s + (0.0534 + 0.0925i)11-s − 0.277·13-s + (0.442 + 0.765i)17-s + (−0.0740 + 0.128i)19-s + (0.288 − 0.499i)21-s + (−0.208 + 0.361i)23-s + 0.192·27-s − 0.611·29-s + (−0.359 − 0.622i)31-s + (0.0308 − 0.0534i)33-s + (−0.188 + 0.326i)37-s + (0.0800 + 0.138i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.094687326\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.094687326\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.32 - 2.29i)T \) |
good | 11 | \( 1 + (-0.177 - 0.306i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 - 3.15i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.322 - 0.559i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1 - 1.73i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.29T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.14 - 1.98i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 5.64T + 41T^{2} \) |
| 43 | \( 1 + 3.29T + 43T^{2} \) |
| 47 | \( 1 + (6.46 - 11.2i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.82 - 6.62i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.468 - 0.811i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.79 + 10.0i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.96 - 6.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.29T + 71T^{2} \) |
| 73 | \( 1 + (-2.14 - 3.71i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.677 + 1.17i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 1.64T + 83T^{2} \) |
| 89 | \( 1 + (3.46 - 6.00i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.70T + 97T^{2} \) |
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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
−9.282646362697000895823726870508, −8.346148937779003071467517044005, −7.87541352334291977191132751531, −6.96536158493199129487044825318, −6.04927993799897177030509708446, −5.48301074512506970548085492341, −4.60012284384306102837471553770, −3.44919644329299652648446655580, −2.29075832846758112812193804957, −1.42745328190714555448014978941,
0.40180186524654997485315870058, 1.81440214378149941329638288945, 3.21087560518290963776558188442, 4.01290068718196905377431338845, 4.93352777486883076971376418313, 5.47886809451890061693728663487, 6.70687128231397452117424563921, 7.23996046771952952315750066326, 8.180855284243402403465425896421, 8.894165583230506450504770602923