L(s) = 1 | + (−1.26 + 2.18i)5-s + (−2.63 + 0.275i)7-s + (2.85 + 4.94i)11-s + (−2.45 − 4.24i)13-s + (−2.49 + 4.32i)17-s + (0.00383 + 0.00664i)19-s + (−0.333 + 0.578i)23-s + (−0.682 − 1.18i)25-s + (−3.85 + 6.66i)29-s + 7.76·31-s + (2.71 − 6.09i)35-s + (−3.19 − 5.53i)37-s + (−5.21 − 9.02i)41-s + (−4.42 + 7.67i)43-s + 2.16·47-s + ⋯ |
L(s) = 1 | + (−0.564 + 0.977i)5-s + (−0.994 + 0.104i)7-s + (0.861 + 1.49i)11-s + (−0.680 − 1.17i)13-s + (−0.605 + 1.04i)17-s + (0.000880 + 0.00152i)19-s + (−0.0696 + 0.120i)23-s + (−0.136 − 0.236i)25-s + (−0.715 + 1.23i)29-s + 1.39·31-s + (0.459 − 1.03i)35-s + (−0.525 − 0.909i)37-s + (−0.813 − 1.40i)41-s + (−0.675 + 1.16i)43-s + 0.315·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1348002306\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1348002306\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.63 - 0.275i)T \) |
good | 5 | \( 1 + (1.26 - 2.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.85 - 4.94i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.45 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.49 - 4.32i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.00383 - 0.00664i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.333 - 0.578i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.85 - 6.66i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.76T + 31T^{2} \) |
| 37 | \( 1 + (3.19 + 5.53i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (5.21 + 9.02i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.42 - 7.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.16T + 47T^{2} \) |
| 53 | \( 1 + (-3.69 + 6.40i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 0.523T + 59T^{2} \) |
| 61 | \( 1 + 8.99T + 61T^{2} \) |
| 67 | \( 1 - 5.09T + 67T^{2} \) |
| 71 | \( 1 + 5.68T + 71T^{2} \) |
| 73 | \( 1 + (1.52 - 2.63i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 6.16T + 79T^{2} \) |
| 83 | \( 1 + (0.258 - 0.448i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.19 + 2.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.32 + 7.49i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.258346205630195304018228416695, −8.458446634396619322571309640613, −7.41210665670210880168981846770, −7.03137103544030425037120290096, −6.40690065380131595356820072478, −5.44772787478103159005024549764, −4.38248058368188151657250405760, −3.58465562045425701876692963851, −2.88452265183293205497621316998, −1.80694016326564937193011977471,
0.04774354698359463056917147684, 1.05049082079088986351640355500, 2.54819638227765813389484447954, 3.53102735102941267361603466858, 4.31651100178169353456816603843, 4.97523966145878015161378142149, 6.17127159332887666190744216652, 6.59043548924063476524039268103, 7.49329085713159561493480856754, 8.469371891615654270467586813626