L(s) = 1 | + (1.11 − 1.11i)2-s + (1.51 − 0.843i)3-s − 0.487i·4-s + (0.746 − 2.62i)6-s + (−0.707 − 0.707i)7-s + (1.68 + 1.68i)8-s + (1.57 − 2.55i)9-s − 2.87i·11-s + (−0.411 − 0.737i)12-s + (1.36 − 1.36i)13-s − 1.57·14-s + 4.73·16-s + (−4.87 + 4.87i)17-s + (−1.08 − 4.60i)18-s − 3.03i·19-s + ⋯ |
L(s) = 1 | + (0.788 − 0.788i)2-s + (0.873 − 0.486i)3-s − 0.243i·4-s + (0.304 − 1.07i)6-s + (−0.267 − 0.267i)7-s + (0.596 + 0.596i)8-s + (0.525 − 0.850i)9-s − 0.865i·11-s + (−0.118 − 0.212i)12-s + (0.378 − 0.378i)13-s − 0.421·14-s + 1.18·16-s + (−1.18 + 1.18i)17-s + (−0.256 − 1.08i)18-s − 0.697i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.18313 - 1.82895i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.18313 - 1.82895i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.51 + 0.843i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (-1.11 + 1.11i)T - 2iT^{2} \) |
| 11 | \( 1 + 2.87iT - 11T^{2} \) |
| 13 | \( 1 + (-1.36 + 1.36i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.87 - 4.87i)T - 17iT^{2} \) |
| 19 | \( 1 + 3.03iT - 19T^{2} \) |
| 23 | \( 1 + (-4.40 - 4.40i)T + 23iT^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 2.57T + 31T^{2} \) |
| 37 | \( 1 + (4.41 + 4.41i)T + 37iT^{2} \) |
| 41 | \( 1 - 10.2iT - 41T^{2} \) |
| 43 | \( 1 + (4.59 - 4.59i)T - 43iT^{2} \) |
| 47 | \( 1 + (6.91 - 6.91i)T - 47iT^{2} \) |
| 53 | \( 1 + (2.23 + 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.66T + 59T^{2} \) |
| 61 | \( 1 - 2.03T + 61T^{2} \) |
| 67 | \( 1 + (-11.0 - 11.0i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.76iT - 71T^{2} \) |
| 73 | \( 1 + (4.85 - 4.85i)T - 73iT^{2} \) |
| 79 | \( 1 + 11.3iT - 79T^{2} \) |
| 83 | \( 1 + (7.16 + 7.16i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.776T + 89T^{2} \) |
| 97 | \( 1 + (1.56 + 1.56i)T + 97iT^{2} \) |
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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
−11.04478245282170296988769726765, −9.877268543220167844839191148741, −8.735857959601727437432538231845, −8.142711977074050826416114048380, −7.04579411643644784296973352319, −6.00120825857639566833437668587, −4.57506894331881874170337143762, −3.54954112587488433886831356028, −2.86889484881180433877899964036, −1.51092874851301032781498242978,
2.08828984464108470204108551375, 3.51988252807989890557112781021, 4.56037413785837609451473093512, 5.20157320729372020511962794194, 6.64386773220444029618201119352, 7.14646839191945518589124035323, 8.361570078404043166834464639438, 9.241609524884490955638535869334, 10.02289504419459374194362044670, 10.88878855160787244100454159833