| L(s) = 1 | + (−0.581 + 1.28i)2-s + (−1.32 − 1.50i)4-s + (2.70 − 0.832i)8-s − 6.57·11-s + (−0.5 + 3.96i)16-s + (3.82 − 8.46i)22-s + 1.91·23-s + 5·25-s + 6.06i·29-s + (−4.82 − 2.95i)32-s + 10.5·37-s − 12i·43-s + (8.69 + 9.85i)44-s + (−1.11 + 2.46i)46-s + (−2.90 + 6.44i)50-s + ⋯ |
| L(s) = 1 | + (−0.411 + 0.911i)2-s + (−0.661 − 0.750i)4-s + (0.955 − 0.294i)8-s − 1.98·11-s + (−0.125 + 0.992i)16-s + (0.815 − 1.80i)22-s + 0.399·23-s + 25-s + 1.12i·29-s + (−0.852 − 0.522i)32-s + 1.73·37-s − 1.82i·43-s + (1.31 + 1.48i)44-s + (−0.164 + 0.363i)46-s + (−0.411 + 0.911i)50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9486225708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9486225708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.581 - 1.28i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 11 | \( 1 + 6.57T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.91T + 23T^{2} \) |
| 29 | \( 1 - 6.06iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 14.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 15.8iT - 67T^{2} \) |
| 71 | \( 1 - 15.0T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 15.8iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.127470304873695750558112799334, −8.403263605110292676118737619869, −7.72788851719810542003935879033, −7.07542041413179536064837328826, −6.17108775345765118397961206106, −5.17994840582657064948861940937, −4.85890105877802911144516236262, −3.40918517280381078706709738677, −2.18229772609466143605242245771, −0.54024605251455427120994492065,
0.914093875373577056563662573168, 2.48131658203164216809990923619, 2.90850486927284505701811177868, 4.26118676617323205969576628230, 4.97827102574294768626031526230, 5.93091109227457013785930700977, 7.24266810488167625291569369492, 7.933925816264200354037753139693, 8.462641192092734430615442110036, 9.531694794915584531535530925395
