L(s) = 1 | + (1.07 + 0.286i)2-s + i·3-s + (−0.667 − 0.385i)4-s + (−3.71 + 0.994i)5-s + (−0.286 + 1.07i)6-s + (−1.35 + 2.27i)7-s + (−2.17 − 2.17i)8-s − 9-s − 4.26·10-s + (4.06 + 4.06i)11-s + (0.385 − 0.667i)12-s + (−2.56 − 2.53i)13-s + (−2.10 + 2.04i)14-s + (−0.994 − 3.71i)15-s + (−0.931 − 1.61i)16-s + (−2.31 + 4.00i)17-s + ⋯ |
L(s) = 1 | + (0.757 + 0.202i)2-s + 0.577i·3-s + (−0.333 − 0.192i)4-s + (−1.66 + 0.444i)5-s + (−0.117 + 0.437i)6-s + (−0.512 + 0.858i)7-s + (−0.767 − 0.767i)8-s − 0.333·9-s − 1.34·10-s + (1.22 + 1.22i)11-s + (0.111 − 0.192i)12-s + (−0.711 − 0.702i)13-s + (−0.562 + 0.546i)14-s + (−0.256 − 0.958i)15-s + (−0.232 − 0.403i)16-s + (−0.560 + 0.970i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159672 + 0.722120i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159672 + 0.722120i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 + (1.35 - 2.27i)T \) |
| 13 | \( 1 + (2.56 + 2.53i)T \) |
good | 2 | \( 1 + (-1.07 - 0.286i)T + (1.73 + i)T^{2} \) |
| 5 | \( 1 + (3.71 - 0.994i)T + (4.33 - 2.5i)T^{2} \) |
| 11 | \( 1 + (-4.06 - 4.06i)T + 11iT^{2} \) |
| 17 | \( 1 + (2.31 - 4.00i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.10 - 1.10i)T + 19iT^{2} \) |
| 23 | \( 1 + (-3.73 + 2.15i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.38 - 4.13i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.993 + 3.70i)T + (-26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (0.653 - 2.44i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (4.40 - 1.18i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.65 - 1.53i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.11 - 7.89i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.36 + 2.37i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.64 - 13.6i)T + (-51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + 3.55iT - 61T^{2} \) |
| 67 | \( 1 + (4.37 - 4.37i)T - 67iT^{2} \) |
| 71 | \( 1 + (7.06 + 1.89i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (-1.11 - 0.298i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.33 + 7.51i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.07 - 1.07i)T + 83iT^{2} \) |
| 89 | \( 1 + (-8.76 - 2.34i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.63 + 6.11i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−12.28311341938968129920347491892, −11.70288522208889753144213894676, −10.43095378489504968549797432885, −9.435712191352113605116010818160, −8.571480912085346656136263343605, −7.23496621856233485148453131946, −6.26055200310060210140705286380, −4.86999554788302017994264573667, −4.08136093391349346687068194113, −3.14395259268304018980950470754,
0.45317814608190531438925918919, 3.26616724146536303392572582138, 3.98203475975269659833791238067, 4.99280882193855596902857092798, 6.66524662114008525612324108351, 7.46756698550798622031826665269, 8.591078725125649580009577483432, 9.276780523020847235794019340957, 11.23618174260502443716667658662, 11.67063744812403488640971066720