L(s) = 1 | + (−0.780 + 1.35i)2-s + (1.28 + 2.21i)3-s + (−0.219 − 0.379i)4-s + (−0.5 + 0.866i)5-s − 4·6-s − 2.43·8-s + (−1.78 + 3.08i)9-s + (−0.780 − 1.35i)10-s + (−1.28 − 2.21i)11-s + (0.561 − 0.972i)12-s + 4.56·13-s − 2.56·15-s + (2.34 − 4.05i)16-s + (2.28 + 3.95i)17-s + (−2.78 − 4.81i)18-s + (−0.561 + 0.972i)19-s + ⋯ |
L(s) = 1 | + (−0.552 + 0.956i)2-s + (0.739 + 1.28i)3-s + (−0.109 − 0.189i)4-s + (−0.223 + 0.387i)5-s − 1.63·6-s − 0.862·8-s + (−0.593 + 1.02i)9-s + (−0.246 − 0.427i)10-s + (−0.386 − 0.668i)11-s + (0.162 − 0.280i)12-s + 1.26·13-s − 0.661·15-s + (0.585 − 1.01i)16-s + (0.553 + 0.958i)17-s + (−0.655 − 1.13i)18-s + (−0.128 + 0.223i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0714596 + 1.12605i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0714596 + 1.12605i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.780 - 1.35i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.28 - 2.21i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (1.28 + 2.21i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.56T + 13T^{2} \) |
| 17 | \( 1 + (-2.28 - 3.95i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.56 + 4.43i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 5.68T + 29T^{2} \) |
| 31 | \( 1 + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12T + 43T^{2} \) |
| 47 | \( 1 + (1.84 - 3.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.56 + 2.70i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.68 + 8.11i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.12 - 5.40i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + (2.12 + 3.67i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.28 + 5.68i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (3.56 - 6.16i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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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
−12.62006744429066798154165904212, −11.17378499313080896299262535080, −10.47011955307194223574827338989, −9.396008727657851714960656143303, −8.489227506168596540044939266351, −8.056450103977964157368157518370, −6.61451015253128499123595867193, −5.57207786291029461823556139101, −3.91144286210323581491708728378, −3.07563018957957872740070176019,
1.09397042824764715285815690555, 2.24856897575602246435458230120, 3.50810304108099994306239516545, 5.54530333484489846066949453816, 6.93269721305817459068678208881, 7.84076835513160051268425121000, 8.833013694295558919258981906456, 9.536763186405128613032534823351, 10.83233322406802633473248965172, 11.67929097214995122706531054297