L(s) = 1 | + (0.667 + 1.15i)5-s + (−1.90 + 1.83i)7-s + (−0.756 + 1.31i)11-s + (−2.58 + 4.48i)13-s + (−0.774 − 1.34i)17-s + (1.25 − 2.16i)19-s + (3.68 + 6.37i)23-s + (1.60 − 2.78i)25-s + (0.0309 + 0.0536i)29-s + 3.84·31-s + (−3.39 − 0.972i)35-s + (−0.281 + 0.487i)37-s + (−4.51 + 7.81i)41-s + (−5.09 − 8.83i)43-s − 9.51·47-s + ⋯ |
L(s) = 1 | + (0.298 + 0.516i)5-s + (−0.719 + 0.694i)7-s + (−0.228 + 0.395i)11-s + (−0.717 + 1.24i)13-s + (−0.187 − 0.325i)17-s + (0.287 − 0.497i)19-s + (0.767 + 1.32i)23-s + (0.321 − 0.557i)25-s + (0.00575 + 0.00996i)29-s + 0.691·31-s + (−0.573 − 0.164i)35-s + (−0.0462 + 0.0801i)37-s + (−0.704 + 1.22i)41-s + (−0.777 − 1.34i)43-s − 1.38·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6409333778\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6409333778\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.90 - 1.83i)T \) |
good | 5 | \( 1 + (-0.667 - 1.15i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.756 - 1.31i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.58 - 4.48i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.774 + 1.34i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.25 + 2.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.68 - 6.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0309 - 0.0536i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.84T + 31T^{2} \) |
| 37 | \( 1 + (0.281 - 0.487i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.51 - 7.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.09 + 8.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 9.51T + 47T^{2} \) |
| 53 | \( 1 + (0.755 + 1.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 3.23T + 61T^{2} \) |
| 67 | \( 1 + 6.93T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.37 + 2.38i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 5.91T + 79T^{2} \) |
| 83 | \( 1 + (-2.80 - 4.85i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.703 - 1.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 + 10.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.257399977681545363211150948968, −8.487127572438050877645234548200, −7.39059669962354215241947877350, −6.80986738568436506797183577199, −6.26706760723486418826328154811, −5.20248294361210544827052155288, −4.60630090611872554440608817876, −3.32062032638712667525191290594, −2.66013878914585136660806809303, −1.71638982685601846014052586001,
0.19907604573212221869333258266, 1.30359057440518970210895709350, 2.77597077607806129164028570514, 3.38437378079658750865493840989, 4.55385225951294334690230237093, 5.20355810743133343630330083480, 6.08019461084635111363431967398, 6.79737368144474693254514315062, 7.65390120265058385172514934610, 8.317060162421387311085046642178