| L(s) = 1 | + 0.296i·3-s + (−2.18 − 0.485i)5-s + 3.46i·7-s + 2.91·9-s − 3.11·11-s − 4.60i·13-s + (0.144 − 0.647i)15-s + 5.49i·17-s − 4.48·19-s − 1.02·21-s − i·23-s + (4.52 + 2.11i)25-s + 1.75i·27-s − 9.19·29-s − 5.89·31-s + ⋯ |
| L(s) = 1 | + 0.171i·3-s + (−0.976 − 0.217i)5-s + 1.30i·7-s + 0.970·9-s − 0.938·11-s − 1.27i·13-s + (0.0372 − 0.167i)15-s + 1.33i·17-s − 1.02·19-s − 0.224·21-s − 0.208i·23-s + (0.905 + 0.423i)25-s + 0.337i·27-s − 1.70·29-s − 1.05·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0417972 + 0.380345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0417972 + 0.380345i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.18 + 0.485i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.296iT - 3T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3.11T + 11T^{2} \) |
| 13 | \( 1 + 4.60iT - 13T^{2} \) |
| 17 | \( 1 - 5.49iT - 17T^{2} \) |
| 19 | \( 1 + 4.48T + 19T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 + 5.89T + 31T^{2} \) |
| 37 | \( 1 + 6.95iT - 37T^{2} \) |
| 41 | \( 1 + 9.03T + 41T^{2} \) |
| 43 | \( 1 - 5.55iT - 43T^{2} \) |
| 47 | \( 1 - 5.48iT - 47T^{2} \) |
| 53 | \( 1 - 2.74iT - 53T^{2} \) |
| 59 | \( 1 + 9.33T + 59T^{2} \) |
| 61 | \( 1 + 1.40T + 61T^{2} \) |
| 67 | \( 1 - 3.49iT - 67T^{2} \) |
| 71 | \( 1 - 4.28T + 71T^{2} \) |
| 73 | \( 1 + 4.92iT - 73T^{2} \) |
| 79 | \( 1 + 2.12T + 79T^{2} \) |
| 83 | \( 1 - 16.0iT - 83T^{2} \) |
| 89 | \( 1 + 11.9T + 89T^{2} \) |
| 97 | \( 1 + 4.37iT - 97T^{2} \) |
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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
−10.68929968896938138917532155889, −9.595226168958733005010241451429, −8.647462180996254382467119972603, −8.041002781000399550607530065179, −7.30443184459368379912695629106, −5.97878144254988785448240980900, −5.26363236497416633775524954072, −4.20536789393233861694088457739, −3.23324062318899175647603202648, −1.92321355145543647751025490044,
0.17375867224342794536475916559, 1.85775040993853610130256576517, 3.46799058075954833452820797245, 4.24336974078984000597971801567, 5.00029259187697095992095989943, 6.67862697738115351740444735125, 7.22742341334710549250802791460, 7.68031654176448680276821158560, 8.814482519018190600408070439052, 9.834297682732603679622129184765
