L(s) = 1 | − 1.72·2-s + (−1.49 + 2.58i)3-s + 0.970·4-s + (−1.96 − 3.40i)5-s + (2.57 − 4.45i)6-s + (0.142 − 0.246i)7-s + 1.77·8-s + (−2.94 − 5.10i)9-s + (3.39 + 5.87i)10-s + (2.19 + 3.80i)11-s + (−1.44 + 2.50i)12-s + (−0.5 − 0.866i)13-s + (−0.245 + 0.425i)14-s + 11.7·15-s − 4.99·16-s + (2.72 − 4.71i)17-s + ⋯ |
L(s) = 1 | − 1.21·2-s + (−0.861 + 1.49i)3-s + 0.485·4-s + (−0.879 − 1.52i)5-s + (1.04 − 1.81i)6-s + (0.0538 − 0.0932i)7-s + 0.627·8-s + (−0.982 − 1.70i)9-s + (1.07 + 1.85i)10-s + (0.661 + 1.14i)11-s + (−0.417 + 0.723i)12-s + (−0.138 − 0.240i)13-s + (−0.0656 + 0.113i)14-s + 3.03·15-s − 1.24·16-s + (0.660 − 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0605 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.227899 + 0.242136i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.227899 + 0.242136i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + (3.07 - 4.64i)T \) |
good | 2 | \( 1 + 1.72T + 2T^{2} \) |
| 3 | \( 1 + (1.49 - 2.58i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (1.96 + 3.40i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.142 + 0.246i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.19 - 3.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.72 + 4.71i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.983 - 1.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 0.503T + 23T^{2} \) |
| 29 | \( 1 + 1.05T + 29T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.42 - 4.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.52 - 4.37i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 13.5T + 47T^{2} \) |
| 53 | \( 1 + (5.72 + 9.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.96 - 6.86i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + (-0.707 - 1.22i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.19 - 9.00i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.53 - 7.85i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.01 - 6.96i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (0.505 + 0.875i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 + 1.31T + 97T^{2} \) |
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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
−11.34809450421054576004113408957, −10.29277052870744008897304493469, −9.560221613320910172652435826989, −9.111279875851753019297291618678, −8.153898157815809901766612119060, −7.12038857992286659062651139945, −5.32156868316412188115481200722, −4.66981466940181045355322697537, −3.93276846963556788523409533059, −0.995734086912398261125855202002,
0.49244628724699744124650724109, 2.08321532946690865333465629388, 3.75429684140737612615178620247, 5.83622508124815262250805457347, 6.65562515507233323837692476286, 7.40166515232848219619873810881, 7.969616733985604297350705631721, 8.964257219508848677872328112674, 10.54127945148458337950118293566, 10.94606218206254508697274951809