L(s) = 1 | + (1.22 + 1.22i)2-s + (−1.22 − 1.22i)3-s + 0.999i·4-s − 2.99i·6-s + (−2.44 + 2.44i)7-s + (1.22 − 1.22i)8-s + 2.99i·9-s − i·11-s + (1.22 − 1.22i)12-s + (2.44 + 2.44i)13-s − 5.99·14-s + 5·16-s + (4.89 + 4.89i)17-s + (−3.67 + 3.67i)18-s − 2i·19-s + ⋯ |
L(s) = 1 | + (0.866 + 0.866i)2-s + (−0.707 − 0.707i)3-s + 0.499i·4-s − 1.22i·6-s + (−0.925 + 0.925i)7-s + (0.433 − 0.433i)8-s + 0.999i·9-s − 0.301i·11-s + (0.353 − 0.353i)12-s + (0.679 + 0.679i)13-s − 1.60·14-s + 1.25·16-s + (1.18 + 1.18i)17-s + (−0.866 + 0.866i)18-s − 0.458i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.41752 + 1.12185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.41752 + 1.12185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + (-1.22 - 1.22i)T + 2iT^{2} \) |
| 7 | \( 1 + (2.44 - 2.44i)T - 7iT^{2} \) |
| 13 | \( 1 + (-2.44 - 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + (-4.89 - 4.89i)T + 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (4.89 - 4.89i)T - 23iT^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 37iT^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 + (-7.34 - 7.34i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.89 + 4.89i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.89 - 4.89i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + (2.44 - 2.44i)T - 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + (2.44 + 2.44i)T + 73iT^{2} \) |
| 79 | \( 1 + 10iT - 79T^{2} \) |
| 83 | \( 1 + (7.34 - 7.34i)T - 83iT^{2} \) |
| 89 | \( 1 + 12T + 89T^{2} \) |
| 97 | \( 1 + (-9.79 + 9.79i)T - 97iT^{2} \) |
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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
−10.42490707393289772862678315055, −9.617554211945326742721644929711, −8.364043689083454534095295860998, −7.60451602090778475902914491636, −6.35342702350562535013859390636, −6.23206132091284541611808403328, −5.48163109173721446746926093859, −4.35099814839084804542718503123, −3.12292974216866090908312791472, −1.43094132361273501017526129662,
0.831867587077564432072177027364, 2.86193901915992400633092457262, 3.67210192696209675670644753896, 4.38556169936754647212739060691, 5.37112422912152037541155954165, 6.24854159547523000432739262837, 7.31397505138621098325041298662, 8.418747054610043074090793811384, 9.887104328873727536242385483887, 10.14340022067073463857904222678