L(s) = 1 | + 2-s + 4-s + (−0.526 + 2.17i)5-s + 8-s + (−0.526 + 2.17i)10-s + 4.41i·11-s + 4.94·13-s + 16-s + 5.22i·17-s − 4.97i·19-s + (−0.526 + 2.17i)20-s + 4.41i·22-s − 7.71·23-s + (−4.44 − 2.28i)25-s + 4.94·26-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + (−0.235 + 0.971i)5-s + 0.353·8-s + (−0.166 + 0.687i)10-s + 1.33i·11-s + 1.37·13-s + 0.250·16-s + 1.26i·17-s − 1.14i·19-s + (−0.117 + 0.485i)20-s + 0.942i·22-s − 1.60·23-s + (−0.889 − 0.457i)25-s + 0.969·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 - 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.529630904\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.529630904\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.526 - 2.17i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4.41iT - 11T^{2} \) |
| 13 | \( 1 - 4.94T + 13T^{2} \) |
| 17 | \( 1 - 5.22iT - 17T^{2} \) |
| 19 | \( 1 + 4.97iT - 19T^{2} \) |
| 23 | \( 1 + 7.71T + 23T^{2} \) |
| 29 | \( 1 - 10.2iT - 29T^{2} \) |
| 31 | \( 1 + 6.43iT - 31T^{2} \) |
| 37 | \( 1 - 9.25iT - 37T^{2} \) |
| 41 | \( 1 - 6.79T + 41T^{2} \) |
| 43 | \( 1 - 2.08iT - 43T^{2} \) |
| 47 | \( 1 + 7.83iT - 47T^{2} \) |
| 53 | \( 1 - 5.67T + 53T^{2} \) |
| 59 | \( 1 + 12.1T + 59T^{2} \) |
| 61 | \( 1 - 6.76iT - 61T^{2} \) |
| 67 | \( 1 + 2.40iT - 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 - 0.202T + 73T^{2} \) |
| 79 | \( 1 + 0.736T + 79T^{2} \) |
| 83 | \( 1 - 3.96iT - 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.418695830892334150119803070353, −7.72285457769605874129804272441, −6.99988096217414222208790696809, −6.35994403658321670958155175905, −5.84667595067014816985154709544, −4.70861294136978975827136980646, −4.02372164175458135226096962223, −3.39042439483822501813978772410, −2.38205976878566154505616559391, −1.55136903439590394940675822047,
0.53337635286742738257586139573, 1.56540803148598772524249948524, 2.76606562182798083476989197515, 3.86049152622999622895342514645, 4.08345124178157151571878533784, 5.25160757818842179410770092662, 5.93011936031106410713707599008, 6.21517702651258494523436583843, 7.58179191901390224010306907641, 8.037907425172817777951706177992