L(s) = 1 | − i·3-s + (−0.539 + 2.17i)5-s − i·7-s − 9-s − 0.539·11-s − 2.24i·13-s + (2.17 + 0.539i)15-s − 3.24i·17-s − 2.24·19-s − 21-s − i·23-s + (−4.41 − 2.34i)25-s + i·27-s + 8.51·29-s − 6.70·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.241 + 0.970i)5-s − 0.377i·7-s − 0.333·9-s − 0.162·11-s − 0.623i·13-s + (0.560 + 0.139i)15-s − 0.787i·17-s − 0.515·19-s − 0.218·21-s − 0.208i·23-s + (−0.883 − 0.468i)25-s + 0.192i·27-s + 1.58·29-s − 1.20·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.086815836\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.086815836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.539 - 2.17i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 0.539T + 11T^{2} \) |
| 13 | \( 1 + 2.24iT - 13T^{2} \) |
| 17 | \( 1 + 3.24iT - 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 7.04iT - 37T^{2} \) |
| 41 | \( 1 - 5.14T + 41T^{2} \) |
| 43 | \( 1 + 4.97iT - 43T^{2} \) |
| 47 | \( 1 + 8.87iT - 47T^{2} \) |
| 53 | \( 1 + 4.80iT - 53T^{2} \) |
| 59 | \( 1 - 0.170T + 59T^{2} \) |
| 61 | \( 1 + 2.67T + 61T^{2} \) |
| 67 | \( 1 + 8.31iT - 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 8.09iT - 73T^{2} \) |
| 79 | \( 1 + 5.57T + 79T^{2} \) |
| 83 | \( 1 - 0.411iT - 83T^{2} \) |
| 89 | \( 1 - 6.68T + 89T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−9.324527269412785750462718811099, −8.388149110098643522922761839408, −7.53696894403896182796415224550, −7.04176226391883817887593943588, −6.21424706339756113769281520767, −5.27850682712706240461614187987, −4.05512352955260696291153529710, −3.06397088074925791257494830908, −2.16794299569408443005522095904, −0.45367968493702258984870384506,
1.40342519190880397215528845339, 2.77792864161789050613355794830, 4.06812720047121556237899066177, 4.60618732892329542147131398640, 5.58516488865258406035333239840, 6.34875831981042300438853456709, 7.56700315741502744625235129816, 8.450208899562433141567351153714, 8.930105506381315095964394945510, 9.704422680060286559147622711129