L(s) = 1 | + (0.965 + 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 + 0.499i)4-s + i·6-s + (2.64 + 0.153i)7-s + (0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (2.27 − 3.94i)11-s + (−0.258 + 0.965i)12-s + (1.77 − 1.77i)13-s + (2.51 + 0.831i)14-s + (0.500 + 0.866i)16-s + (3.98 − 1.06i)17-s + (−0.965 + 0.258i)18-s + (−1.88 − 3.27i)19-s + ⋯ |
L(s) = 1 | + (0.683 + 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 + 0.249i)4-s + 0.408i·6-s + (0.998 + 0.0579i)7-s + (0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.686 − 1.18i)11-s + (−0.0747 + 0.278i)12-s + (0.493 − 0.493i)13-s + (0.671 + 0.222i)14-s + (0.125 + 0.216i)16-s + (0.966 − 0.258i)17-s + (−0.227 + 0.0610i)18-s + (−0.433 − 0.750i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.991354591\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.991354591\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.64 - 0.153i)T \) |
good | 11 | \( 1 + (-2.27 + 3.94i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.77 + 1.77i)T - 13iT^{2} \) |
| 17 | \( 1 + (-3.98 + 1.06i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.88 + 3.27i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.08 - 7.77i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 1.55iT - 29T^{2} \) |
| 31 | \( 1 + (-3.37 - 1.94i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (11.0 + 2.95i)T + (32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 11.3iT - 41T^{2} \) |
| 43 | \( 1 + (0.367 + 0.367i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.30 - 4.87i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.14 - 2.18i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-0.221 + 0.383i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.09 + 4.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.41 + 8.99i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 6.68T + 71T^{2} \) |
| 73 | \( 1 + (-1.12 - 4.20i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.08 + 2.35i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.21 - 3.21i)T - 83iT^{2} \) |
| 89 | \( 1 + (3.02 + 5.23i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.462 - 0.462i)T + 97iT^{2} \) |
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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
−10.07584515394297445875663306633, −9.052560272929101723453967693488, −8.274873469377541030403939437200, −7.62058446252961570668793851316, −6.36493730044988504298224135423, −5.56169740054855629714208358342, −4.83909698005433956816892671041, −3.74930715931906250385446824385, −3.04215889288583402830744942870, −1.41845302071805948534261963659,
1.45263513121864355692270095312, 2.18865362109972699556429753446, 3.71973641552967073631567610726, 4.47878288242061646220281144741, 5.47249062902824872681562972381, 6.49876308279162936036324448523, 7.16134491717770433070610625302, 8.144535812832115405772909399960, 8.816666089892466407341842699558, 10.09251193700476859824901141744