L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (1.80 + 1.31i)5-s + (−0.809 − 0.587i)6-s + 2·7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.690 − 2.12i)10-s + (−1.61 − 4.97i)11-s + (−0.309 + 0.951i)12-s + (−1.5 + 4.61i)13-s + (−0.618 − 1.90i)14-s + 2.23·15-s + (0.309 − 0.951i)16-s + (−6.35 − 4.61i)17-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (0.467 − 0.339i)3-s + (−0.404 + 0.293i)4-s + (0.809 + 0.587i)5-s + (−0.330 − 0.239i)6-s + 0.755·7-s + (0.286 + 0.207i)8-s + (0.103 − 0.317i)9-s + (0.218 − 0.672i)10-s + (−0.487 − 1.50i)11-s + (−0.0892 + 0.274i)12-s + (−0.416 + 1.28i)13-s + (−0.165 − 0.508i)14-s + 0.577·15-s + (0.0772 − 0.237i)16-s + (−1.54 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13165 - 0.532515i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13165 - 0.532515i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (-1.80 - 1.31i)T \) |
good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + (1.61 + 4.97i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.5 - 4.61i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (6.35 + 4.61i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.23 - 1.62i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.85 - 5.70i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-1.11 + 0.812i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (3 + 2.17i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.663 - 2.04i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.88 - 5.79i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 + (3.85 - 2.80i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (6.92 - 5.03i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.76 + 8.50i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.73 + 8.42i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.85 - 5.70i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-11.4 + 8.33i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.972 + 2.99i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.85 + 3.52i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.427 - 1.31i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-11.2 + 8.14i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.10961297046229612612792468592, −11.34184022606453389640632226398, −11.28851132059886494506383363423, −9.687818995382257380037828020586, −8.967736304019538588441462549927, −7.76483694790135152754973126685, −6.52182546105273119245285082556, −4.98102964864389107454151966734, −3.16269405794208447701174193661, −1.89545622112054634888781633824,
2.13936383242117023427396859604, 4.56622908050813456726919366037, 5.28497309654397536401566236720, 6.86851847503758559561602606013, 8.095332281353903721876412967852, 8.881303827775368396817517740481, 9.991425077666941217757049930767, 10.70836891759811505260094371003, 12.61384076149909496095755422090, 13.13954061363547768375896023128