L(s) = 1 | + (−0.927 + 1.06i)2-s + (−0.280 − 1.98i)4-s + (−2.18 + 0.468i)5-s + 3.02·7-s + (2.37 + 1.53i)8-s + (1.52 − 2.76i)10-s − 3.62i·11-s − 1.69·13-s + (−2.80 + 3.22i)14-s + (−3.84 + 1.11i)16-s + 6.60·17-s + 5.12·19-s + (1.54 + 4.19i)20-s + (3.86 + 3.35i)22-s + 6.67i·23-s + ⋯ |
L(s) = 1 | + (−0.655 + 0.755i)2-s + (−0.140 − 0.990i)4-s + (−0.977 + 0.209i)5-s + 1.14·7-s + (0.839 + 0.543i)8-s + (0.482 − 0.875i)10-s − 1.09i·11-s − 0.470·13-s + (−0.748 + 0.862i)14-s + (−0.960 + 0.277i)16-s + 1.60·17-s + 1.17·19-s + (0.344 + 0.938i)20-s + (0.824 + 0.716i)22-s + 1.39i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.899978 + 0.274530i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.899978 + 0.274530i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.927 - 1.06i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.18 - 0.468i)T \) |
good | 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 + 3.62iT - 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 - 6.67iT - 23T^{2} \) |
| 29 | \( 1 - 6.82T + 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 0.371T + 37T^{2} \) |
| 41 | \( 1 + 5.83iT - 41T^{2} \) |
| 43 | \( 1 - 5.24iT - 43T^{2} \) |
| 47 | \( 1 - 0.525iT - 47T^{2} \) |
| 53 | \( 1 + 10.0iT - 53T^{2} \) |
| 59 | \( 1 + 4.86iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.1iT - 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 9.33iT - 97T^{2} \) |
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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
−11.46643039670962436241423630081, −10.58321700141248259227191831213, −9.575761158437981912849652108490, −8.367009637623542537937766511235, −7.87471890001156100151517763094, −7.13137381441613030626373567921, −5.68190158765110080450134605241, −4.88771550797976371629757822555, −3.34179347760667771702755877336, −1.11679228278626990067883806150,
1.19750912932459366113184230362, 2.83224728154688256357323001915, 4.24696552037525332265410644414, 5.02195608828435059608801943733, 7.13677281868438034796750738639, 7.80835007963961983626288405649, 8.458110893960455883626628440740, 9.664166194691945751904503705308, 10.43900077134047876360701154283, 11.44035343959092763033559846999