L(s) = 1 | + (−0.0732 − 0.126i)2-s + (0.157 − 1.72i)3-s + (0.989 − 1.71i)4-s + 2.57·5-s + (−0.230 + 0.106i)6-s + (−1.73 + 3.01i)7-s − 0.582·8-s + (−2.95 − 0.543i)9-s + (−0.188 − 0.326i)10-s + (2.07 − 3.60i)11-s + (−2.79 − 1.97i)12-s + (−2.29 + 3.98i)13-s + 0.509·14-s + (0.404 − 4.43i)15-s + (−1.93 − 3.35i)16-s + (−1.50 + 2.61i)17-s + ⋯ |
L(s) = 1 | + (−0.0518 − 0.0897i)2-s + (0.0908 − 0.995i)3-s + (0.494 − 0.856i)4-s + 1.14·5-s + (−0.0940 + 0.0434i)6-s + (−0.657 + 1.13i)7-s − 0.206·8-s + (−0.983 − 0.181i)9-s + (−0.0595 − 0.103i)10-s + (0.626 − 1.08i)11-s + (−0.808 − 0.570i)12-s + (−0.637 + 1.10i)13-s + 0.136·14-s + (0.104 − 1.14i)15-s + (−0.483 − 0.838i)16-s + (−0.365 + 0.633i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10144 - 0.788632i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10144 - 0.788632i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.157 + 1.72i)T \) |
| 19 | \( 1 + (-3.65 - 2.37i)T \) |
good | 2 | \( 1 + (0.0732 + 0.126i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 2.57T + 5T^{2} \) |
| 7 | \( 1 + (1.73 - 3.01i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.07 + 3.60i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.29 - 3.98i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.50 - 2.61i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2.45 + 4.25i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3.71T + 29T^{2} \) |
| 31 | \( 1 + (-3.31 - 5.75i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.28T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 43 | \( 1 + (2.39 + 4.14i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.71T + 47T^{2} \) |
| 53 | \( 1 + (-5.35 - 9.27i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 - 0.944T + 61T^{2} \) |
| 67 | \( 1 + (0.688 - 1.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.45 + 7.71i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.14 - 3.71i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.69 + 2.93i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.52 + 7.83i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.96 + 6.86i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.930 + 1.61i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.42002070611476717783028074970, −11.78830067774959896604581969329, −10.59521524564033128234788394349, −9.322098454359142896668991040758, −8.839114393447591674291120528324, −6.92231462738879812049578221862, −6.14042051413142075949608931080, −5.56606744009764474886291569487, −2.75610333043529341200519466484, −1.66155640910534132401050105198,
2.67047100413379607051051434545, 3.91605895325183083578901973633, 5.31772834375181186786462443439, 6.77050694628980325777263766757, 7.65597655942940894256597168302, 9.357555265876191776636721665737, 9.769231715995523333626004355430, 10.77547905961646670563530712057, 11.90608257964112049335563586250, 13.14473163019632781527981615756