L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (1.62 − 2.09i)7-s + 0.999i·8-s + (2.59 − 1.5i)11-s − 2.44i·13-s + (2.44 − 0.999i)14-s + (−0.5 + 0.866i)16-s + (−0.507 − 0.878i)17-s + (−0.878 − 0.507i)19-s + 3·22-s + (−3.67 − 2.12i)23-s + (1.22 − 2.12i)26-s + (2.62 + 0.358i)28-s − 1.24i·29-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (0.612 − 0.790i)7-s + 0.353i·8-s + (0.783 − 0.452i)11-s − 0.679i·13-s + (0.654 − 0.267i)14-s + (−0.125 + 0.216i)16-s + (−0.123 − 0.213i)17-s + (−0.201 − 0.116i)19-s + 0.639·22-s + (−0.766 − 0.442i)23-s + (0.240 − 0.416i)26-s + (0.495 + 0.0677i)28-s − 0.230i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.780289142\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.780289142\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.62 + 2.09i)T \) |
good | 11 | \( 1 + (-2.59 + 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.44iT - 13T^{2} \) |
| 17 | \( 1 + (0.507 + 0.878i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.878 + 0.507i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.67 + 2.12i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 1.24iT - 29T^{2} \) |
| 31 | \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.12 + 7.13i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2.02T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 + (0.507 - 0.878i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.07 + 0.621i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.2iT - 71T^{2} \) |
| 73 | \( 1 + (7.24 - 4.18i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.62 + 9.73i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 3.76iT - 97T^{2} \) |
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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
−8.347409702128952980881043253731, −7.83703031238422927227934126385, −7.03538933370464312075480138751, −6.30055599212651985442746816882, −5.59070189509363875684863529177, −4.61403630259292776657960119780, −4.05903620542300177525420504881, −3.18058444351203536781635439010, −2.02829338937267211144962754552, −0.70558779521651691287186195459,
1.43011688794251632095660502580, 2.11719480203679649183899570274, 3.19470665456598952431332159511, 4.19322323973446216009098073691, 4.77741234492252665186400252493, 5.64345102024019245175992096971, 6.43600298041568673007329371329, 7.03927178971431411878170296323, 8.201407846138360551342858366019, 8.677344708707020723745881790432