L(s) = 1 | + 2-s + (0.400 + 1.68i)3-s + 4-s + (0.5 − 0.866i)5-s + (0.400 + 1.68i)6-s + (0.0665 − 2.64i)7-s + 8-s + (−2.67 + 1.35i)9-s + (0.5 − 0.866i)10-s + (2.14 + 3.71i)11-s + (0.400 + 1.68i)12-s + (2.19 + 3.80i)13-s + (0.0665 − 2.64i)14-s + (1.65 + 0.495i)15-s + 16-s + (2.45 − 4.25i)17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (0.231 + 0.972i)3-s + 0.5·4-s + (0.223 − 0.387i)5-s + (0.163 + 0.687i)6-s + (0.0251 − 0.999i)7-s + 0.353·8-s + (−0.892 + 0.450i)9-s + (0.158 − 0.273i)10-s + (0.646 + 1.12i)11-s + (0.115 + 0.486i)12-s + (0.608 + 1.05i)13-s + (0.0177 − 0.706i)14-s + (0.428 + 0.127i)15-s + 0.250·16-s + (0.595 − 1.03i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.43828 + 0.928212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.43828 + 0.928212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.400 - 1.68i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.0665 + 2.64i)T \) |
good | 11 | \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.19 - 3.80i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.45 + 4.25i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.97 - 3.42i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.888 - 1.53i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.79 + 4.84i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.53T + 31T^{2} \) |
| 37 | \( 1 + (-2.40 - 4.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.29 + 3.96i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.90 + 8.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + (4.18 - 7.25i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 5.47T + 61T^{2} \) |
| 67 | \( 1 - 9.14T + 67T^{2} \) |
| 71 | \( 1 + 4.03T + 71T^{2} \) |
| 73 | \( 1 + (-4.69 + 8.12i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 1.10T + 79T^{2} \) |
| 83 | \( 1 + (-3.33 + 5.77i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (6.54 + 11.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.883 + 1.53i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.66137459599749964332624979805, −9.762465343220392178991702477252, −9.318387358045939620132996453460, −7.992384816628024387929018019995, −7.08198947462820132299543910061, −6.00608542469795718804316933263, −4.83182783378024134292696487326, −4.24103444621365293637316639854, −3.36718704605489796478750716701, −1.70938659062537190550455283392,
1.39229629365785063434520389982, 2.83972434168671100782123792341, 3.44896802652673114692159941694, 5.30737675289227279210364319785, 6.06641024620013946081761354724, 6.57612703383732396354436527369, 7.924599444543584601434326196075, 8.483689971098310743976772509878, 9.514949315157431815152118728757, 10.96057900759304829878911505532