| L(s) = 1 | + (0.629 + 1.26i)2-s + (−1.20 + 1.59i)4-s − 2.32i·5-s + (−2.77 − 0.524i)8-s + (2.94 − 1.46i)10-s + 3.58i·11-s − 2.93i·13-s + (−1.08 − 3.84i)16-s − 2.32i·17-s − 8.33·19-s + (3.71 + 2.80i)20-s + (−4.53 + 2.25i)22-s + 1.48i·23-s − 0.414·25-s + (3.71 − 1.84i)26-s + ⋯ |
| L(s) = 1 | + (0.445 + 0.895i)2-s + (−0.603 + 0.797i)4-s − 1.04i·5-s + (−0.982 − 0.185i)8-s + (0.931 − 0.463i)10-s + 1.07i·11-s − 0.812i·13-s + (−0.271 − 0.962i)16-s − 0.564i·17-s − 1.91·19-s + (0.829 + 0.628i)20-s + (−0.966 + 0.480i)22-s + 0.309i·23-s − 0.0828·25-s + (0.727 − 0.361i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4082443653\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4082443653\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.629 - 1.26i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 2.32iT - 5T^{2} \) |
| 11 | \( 1 - 3.58iT - 11T^{2} \) |
| 13 | \( 1 + 2.93iT - 13T^{2} \) |
| 17 | \( 1 + 2.32iT - 17T^{2} \) |
| 19 | \( 1 + 8.33T + 19T^{2} \) |
| 23 | \( 1 - 1.48iT - 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 3.45T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 - 2.89iT - 41T^{2} \) |
| 43 | \( 1 + 6.37iT - 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.59T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 2.93iT - 61T^{2} \) |
| 67 | \( 1 + 9.02iT - 67T^{2} \) |
| 71 | \( 1 - 10.7iT - 71T^{2} \) |
| 73 | \( 1 + 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 15.3iT - 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 13.5iT - 89T^{2} \) |
| 97 | \( 1 + 5.35iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.828913197543262106225961793788, −8.276226305101177095761792810540, −7.44736124022621929320759528481, −6.69528748430994335583095451804, −5.74448875464127994912834489274, −4.93022500417526827088284054695, −4.43657534319352407734522264689, −3.36508178558432113262837476947, −1.94626271765675220942288039642, −0.12285932077456366592653348894,
1.71027101250864604033837361318, 2.65304858274090204557959378697, 3.58436698889537040021688145056, 4.26692688575319013095584923134, 5.42870758181234802767870704713, 6.37238460163971912622400895192, 6.75422201029676381678164145769, 8.235274828103559691629058640727, 8.777941902005951687554046905889, 9.763690888125158488866451633390
