L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1.20 − 2.09i)3-s + (−0.499 − 0.866i)4-s + (−0.292 + 0.507i)5-s + 2.41·6-s + 0.999·8-s + (−1.41 + 2.44i)9-s + (−0.292 − 0.507i)10-s + (−0.5 − 0.866i)11-s + (−1.20 + 2.09i)12-s + 3.82·13-s + 1.41·15-s + (−0.5 + 0.866i)16-s + (1.82 + 3.16i)17-s + (−1.41 − 2.44i)18-s + (−0.292 + 0.507i)19-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.696 − 1.20i)3-s + (−0.249 − 0.433i)4-s + (−0.130 + 0.226i)5-s + 0.985·6-s + 0.353·8-s + (−0.471 + 0.816i)9-s + (−0.0926 − 0.160i)10-s + (−0.150 − 0.261i)11-s + (−0.348 + 0.603i)12-s + 1.06·13-s + 0.365·15-s + (−0.125 + 0.216i)16-s + (0.443 + 0.768i)17-s + (−0.333 − 0.577i)18-s + (−0.0671 + 0.116i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9278820641\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9278820641\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 3 | \( 1 + (1.20 + 2.09i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (0.292 - 0.507i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 3.82T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 - 3.16i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.292 - 0.507i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.12 + 5.40i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.70 + 8.15i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 5.41T + 41T^{2} \) |
| 43 | \( 1 + 5.65T + 43T^{2} \) |
| 47 | \( 1 + (5.24 - 9.08i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.94 + 6.84i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.79 + 4.83i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.91 + 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.37 + 2.38i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + (4.70 + 8.15i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.62 + 11.4i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + (-6.24 + 10.8i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−9.599912921802389419726505640466, −8.625613493641034964049706320499, −7.896595722433195307927465709286, −7.20918272536583867141327357672, −6.21584839128916204335007498033, −6.03924451627818579422295235550, −4.79049274183026592033559907792, −3.40756788457424491678596021631, −1.78676092649032180415758147830, −0.63712503260384284201440131060,
1.13214929686437944892638840308, 2.91865026607353780100327605550, 3.87862954999686439472186788093, 4.75686180566901043375997522135, 5.44564494912569839209184887590, 6.59433945383640406499528540087, 7.75098543146084733288471574219, 8.716234895246902383226557494032, 9.399526193547921689288861321950, 10.17954017375974502192802743617