L(s) = 1 | + (−0.653 − 0.653i)2-s + (−0.707 − 0.707i)3-s − 1.14i·4-s + 0.924i·6-s + (0.741 + 2.53i)7-s + (−2.05 + 2.05i)8-s + 1.00i·9-s − 0.746·11-s + (−0.809 + 0.809i)12-s + (−4.26 − 4.26i)13-s + (1.17 − 2.14i)14-s + 0.398·16-s + (−5.29 + 5.29i)17-s + (0.653 − 0.653i)18-s − 1.17·19-s + ⋯ |
L(s) = 1 | + (−0.462 − 0.462i)2-s + (−0.408 − 0.408i)3-s − 0.572i·4-s + 0.377i·6-s + (0.280 + 0.959i)7-s + (−0.726 + 0.726i)8-s + 0.333i·9-s − 0.225·11-s + (−0.233 + 0.233i)12-s + (−1.18 − 1.18i)13-s + (0.314 − 0.573i)14-s + 0.0996·16-s + (−1.28 + 1.28i)17-s + (0.154 − 0.154i)18-s − 0.269·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.100980 + 0.117721i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100980 + 0.117721i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.741 - 2.53i)T \) |
good | 2 | \( 1 + (0.653 + 0.653i)T + 2iT^{2} \) |
| 11 | \( 1 + 0.746T + 11T^{2} \) |
| 13 | \( 1 + (4.26 + 4.26i)T + 13iT^{2} \) |
| 17 | \( 1 + (5.29 - 5.29i)T - 17iT^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + (3.19 - 3.19i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.45iT - 29T^{2} \) |
| 31 | \( 1 - 4.87iT - 31T^{2} \) |
| 37 | \( 1 + (-2.05 - 2.05i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (-6.64 + 6.64i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.29 + 5.29i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.89 - 4.89i)T - 53iT^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2.52iT - 61T^{2} \) |
| 67 | \( 1 + (6.47 + 6.47i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 + (2.47 + 2.47i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.83iT - 79T^{2} \) |
| 83 | \( 1 + (-0.768 - 0.768i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.69T + 89T^{2} \) |
| 97 | \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \) |
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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
−10.91538118666965092427007805885, −10.40853242113916597131684120991, −9.418509250106809317325821922125, −8.528700101749762151503816459030, −7.69979390887276017422219964982, −6.29401373709057344120163630231, −5.64869003322377715884182544054, −4.70668071862849210438411243346, −2.71231817539502824723049713258, −1.77208999439373727204221284685,
0.10419376464684427957214884108, 2.53407248013544587224358591277, 4.16331943965716645924973429754, 4.66822745725875087877357313725, 6.30061416299411384205077773746, 7.13775203402436335739332947309, 7.70451189315506870399127916568, 9.007113364464341454037788968656, 9.508879086233057664115555446483, 10.58345035573634144895374359329