L(s) = 1 | + (−1.30 + 2.26i)2-s − 2.61·3-s + (−2.42 − 4.20i)4-s + (−1.30 − 2.26i)5-s + (3.42 − 5.93i)6-s + 7.47·8-s + 3.85·9-s + 6.85·10-s + 1.85·11-s + (6.35 + 11.0i)12-s + (−2.5 − 2.59i)13-s + (3.42 + 5.93i)15-s + (−4.92 + 8.53i)16-s + (0.736 + 1.27i)17-s + (−5.04 + 8.73i)18-s − 1.85·19-s + ⋯ |
L(s) = 1 | + (−0.925 + 1.60i)2-s − 1.51·3-s + (−1.21 − 2.10i)4-s + (−0.585 − 1.01i)5-s + (1.39 − 2.42i)6-s + 2.64·8-s + 1.28·9-s + 2.16·10-s + 0.559·11-s + (1.83 + 3.17i)12-s + (−0.693 − 0.720i)13-s + (0.884 + 1.53i)15-s + (−1.23 + 2.13i)16-s + (0.178 + 0.309i)17-s + (−1.18 + 2.05i)18-s − 0.425·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.5 + 2.59i)T \) |
good | 2 | \( 1 + (1.30 - 2.26i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + (1.30 + 2.26i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 17 | \( 1 + (-0.736 - 1.27i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 1.85T + 19T^{2} \) |
| 23 | \( 1 + (-2.23 + 3.87i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.54 + 6.14i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.35 + 4.07i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.381 + 0.661i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.28 - 10.8i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.11 - 1.93i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.88 - 3.25i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.11 - 1.93i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + (-7.09 + 12.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.70T + 83T^{2} \) |
| 89 | \( 1 + (2.45 - 4.25i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.42 - 16.3i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.04060797145740518792909287205, −9.217576369505501305152041262604, −8.242991401730279972782828780390, −7.62520217637335834017986954306, −6.50999378888106660827924594467, −5.97710591948254184475290006370, −4.97042761414765212222222217190, −4.47283315213456736615072293224, −1.00929791706032927283141065071, 0,
1.61714557940711332385785988550, 3.11221270724253022948343790276, 4.09147235014513230646258640690, 5.24514147000313573403109994682, 6.83450220255063357740806204950, 7.27421483754488196004475840244, 8.663908478121610016983128225278, 9.588013591285614976467569094392, 10.43243036520273762396641914601