L(s) = 1 | − 1.93·2-s + (−1.34 − 1.09i)3-s + 1.74·4-s − 3.21i·5-s + (2.59 + 2.12i)6-s + 0.254·7-s + 0.491·8-s + (0.594 + 2.94i)9-s + 6.23i·10-s − 5.68·11-s + (−2.34 − 1.91i)12-s − 2.66i·13-s − 0.491·14-s + (−3.53 + 4.31i)15-s − 4.44·16-s + 4.03i·17-s + ⋯ |
L(s) = 1 | − 1.36·2-s + (−0.774 − 0.633i)3-s + 0.872·4-s − 1.43i·5-s + (1.05 + 0.866i)6-s + 0.0960·7-s + 0.173·8-s + (0.198 + 0.980i)9-s + 1.97i·10-s − 1.71·11-s + (−0.675 − 0.552i)12-s − 0.738i·13-s − 0.131·14-s + (−0.911 + 1.11i)15-s − 1.11·16-s + 0.979i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0250473 + 0.164212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0250473 + 0.164212i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.34 + 1.09i)T \) |
| 59 | \( 1 + (7.12 - 2.86i)T \) |
good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 + 3.21iT - 5T^{2} \) |
| 7 | \( 1 - 0.254T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + 2.66iT - 13T^{2} \) |
| 17 | \( 1 - 4.03iT - 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 - 3.56iT - 29T^{2} \) |
| 31 | \( 1 - 7.81iT - 31T^{2} \) |
| 37 | \( 1 + 5.15iT - 37T^{2} \) |
| 41 | \( 1 + 7.25iT - 41T^{2} \) |
| 43 | \( 1 + 9.79iT - 43T^{2} \) |
| 47 | \( 1 - 5.06T + 47T^{2} \) |
| 53 | \( 1 - 1.16iT - 53T^{2} \) |
| 61 | \( 1 - 0.676iT - 61T^{2} \) |
| 67 | \( 1 - 2.66iT - 67T^{2} \) |
| 71 | \( 1 + 16.2iT - 71T^{2} \) |
| 73 | \( 1 + 13.1iT - 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 + 9.91T + 83T^{2} \) |
| 89 | \( 1 - 7.95T + 89T^{2} \) |
| 97 | \( 1 + 4.47iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.29047589582257049487408185201, −10.65547283448789191658143575265, −10.42175789752673388561630437532, −8.831689904844680518105288663144, −8.179177154598961222650854187464, −7.39223887242312872742745910537, −5.71879654441117092135157641968, −4.79549447942568062278911336079, −1.80939368130398762733752978900, −0.25028104011450143750063096256,
2.62906071688744704892780818425, 4.55846289460673899735296373355, 6.14658450996326248975696372143, 7.19328153956398776061272954335, 8.137538188671499054736771187762, 9.713439522133988979126753756812, 10.06632599297428783908253146625, 11.10190229511788420708012132897, 11.45195241698577571222353149920, 13.19202070856578911557366925179