| L(s) = 1 | + 1.76i·3-s + (−0.432 + 2.19i)5-s − 0.103·9-s − 0.626·11-s + 5.49i·13-s + (−3.86 − 0.761i)15-s − 0.896i·17-s + 6.38·19-s + 3.72i·23-s + (−4.62 − 1.89i)25-s + 5.10i·27-s + 7.87·29-s − 7.52·31-s − 1.10i·33-s + 6i·37-s + ⋯ |
| L(s) = 1 | + 1.01i·3-s + (−0.193 + 0.981i)5-s − 0.0343·9-s − 0.188·11-s + 1.52i·13-s + (−0.997 − 0.196i)15-s − 0.217i·17-s + 1.46·19-s + 0.777i·23-s + (−0.925 − 0.379i)25-s + 0.982i·27-s + 1.46·29-s − 1.35·31-s − 0.192i·33-s + 0.986i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 - 0.193i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.476214354\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.476214354\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.432 - 2.19i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 1.76iT - 3T^{2} \) |
| 11 | \( 1 + 0.626T + 11T^{2} \) |
| 13 | \( 1 - 5.49iT - 13T^{2} \) |
| 17 | \( 1 + 0.896iT - 17T^{2} \) |
| 19 | \( 1 - 6.38T + 19T^{2} \) |
| 23 | \( 1 - 3.72iT - 23T^{2} \) |
| 29 | \( 1 - 7.87T + 29T^{2} \) |
| 31 | \( 1 + 7.52T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 + 7.72T + 41T^{2} \) |
| 43 | \( 1 + 1.72iT - 43T^{2} \) |
| 47 | \( 1 + 5.87iT - 47T^{2} \) |
| 53 | \( 1 - 6.77iT - 53T^{2} \) |
| 59 | \( 1 + 0.593T + 59T^{2} \) |
| 61 | \( 1 + 7.13T + 61T^{2} \) |
| 67 | \( 1 + 5.79iT - 67T^{2} \) |
| 71 | \( 1 - 5.52T + 71T^{2} \) |
| 73 | \( 1 + 3.72iT - 73T^{2} \) |
| 79 | \( 1 - 5.67T + 79T^{2} \) |
| 83 | \( 1 + 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 10.1iT - 97T^{2} \) |
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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
−9.514310513959025150918648608895, −9.120106241690792902136013758771, −7.906052650534431385006974395578, −7.12011500403567622931590409808, −6.53521200719739093047142583727, −5.36792278425932070965759896631, −4.61426129428016841665820901654, −3.69632247988948049469051818247, −3.06578705105591175441893540715, −1.71003283218361743882007571138,
0.56404672460646161349956903855, 1.41824341864306792320228073102, 2.68340825366249047040082410771, 3.77113287934573649680881463495, 4.96392512716299920090701004348, 5.53717025434236375101178193724, 6.50791152679744627337928240788, 7.45177511649329759667478171576, 7.955250163329426069563685407275, 8.564539215562196648354440817514
