L(s) = 1 | + (−0.469 + 0.469i)2-s + 1.55i·4-s + (−1.73 + 1.40i)5-s + (−0.563 − 0.563i)7-s + (−1.67 − 1.67i)8-s + (0.155 − 1.47i)10-s − 5.21i·11-s + (1.60 − 1.60i)13-s + 0.529·14-s − 1.54·16-s + (−2.60 + 2.60i)17-s + i·19-s + (−2.19 − 2.70i)20-s + (2.44 + 2.44i)22-s + (−5.91 − 5.91i)23-s + ⋯ |
L(s) = 1 | + (−0.332 + 0.332i)2-s + 0.779i·4-s + (−0.777 + 0.629i)5-s + (−0.212 − 0.212i)7-s + (−0.590 − 0.590i)8-s + (0.0492 − 0.467i)10-s − 1.57i·11-s + (0.443 − 0.443i)13-s + 0.141·14-s − 0.386·16-s + (−0.632 + 0.632i)17-s + 0.229i·19-s + (−0.490 − 0.605i)20-s + (0.521 + 0.521i)22-s + (−1.23 − 1.23i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.579 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.525776 - 0.271184i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.525776 - 0.271184i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.73 - 1.40i)T \) |
| 19 | \( 1 - iT \) |
good | 2 | \( 1 + (0.469 - 0.469i)T - 2iT^{2} \) |
| 7 | \( 1 + (0.563 + 0.563i)T + 7iT^{2} \) |
| 11 | \( 1 + 5.21iT - 11T^{2} \) |
| 13 | \( 1 + (-1.60 + 1.60i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.60 - 2.60i)T - 17iT^{2} \) |
| 23 | \( 1 + (5.91 + 5.91i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.99T + 29T^{2} \) |
| 31 | \( 1 - 3.10T + 31T^{2} \) |
| 37 | \( 1 + (-3.21 - 3.21i)T + 37iT^{2} \) |
| 41 | \( 1 - 1.54iT - 41T^{2} \) |
| 43 | \( 1 + (-1.72 + 1.72i)T - 43iT^{2} \) |
| 47 | \( 1 + (-1.44 + 1.44i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.07 + 1.07i)T + 53iT^{2} \) |
| 59 | \( 1 + 8.52T + 59T^{2} \) |
| 61 | \( 1 - 5.80T + 61T^{2} \) |
| 67 | \( 1 + (9.57 + 9.57i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.02iT - 71T^{2} \) |
| 73 | \( 1 + (4.66 - 4.66i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.66iT - 79T^{2} \) |
| 83 | \( 1 + (7.43 + 7.43i)T + 83iT^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + (4.26 + 4.26i)T + 97iT^{2} \) |
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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
−10.20693281422304317323516429284, −8.786392117106296588252202898676, −8.296098884128545974602993977859, −7.74201881370958341855403234394, −6.48588244626361790621441881593, −6.20306605949137062048672465217, −4.37435208766830186450642776086, −3.54967779842796836357620725394, −2.77382619975009008184771550218, −0.34211659742797087130851258658,
1.33033053420529489485605792709, 2.52120638136829298684855240179, 4.16716114680626238702854134359, 4.82422967490967668883564283917, 5.91225074090599924019063642255, 6.94193526248520288291099852422, 7.83361510660925364440907102498, 8.894739597041079300492713118752, 9.463930197637903349029861561845, 10.15377218755989322195632473123