| L(s) = 1 | + i·3-s − i·5-s − 7-s − 9-s − 1.08i·11-s − 1.74i·13-s + 15-s − 3.69·17-s + 8.26i·19-s − i·21-s + 7.69·23-s − 25-s − i·27-s − 6.35i·29-s − 3.55·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.377·7-s − 0.333·9-s − 0.326i·11-s − 0.484i·13-s + 0.258·15-s − 0.896·17-s + 1.89i·19-s − 0.218i·21-s + 1.60·23-s − 0.200·25-s − 0.192i·27-s − 1.18i·29-s − 0.637·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.049573073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.049573073\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 1.08iT - 11T^{2} \) |
| 13 | \( 1 + 1.74iT - 13T^{2} \) |
| 17 | \( 1 + 3.69T + 17T^{2} \) |
| 19 | \( 1 - 8.26iT - 19T^{2} \) |
| 23 | \( 1 - 7.69T + 23T^{2} \) |
| 29 | \( 1 + 6.35iT - 29T^{2} \) |
| 31 | \( 1 + 3.55T + 31T^{2} \) |
| 37 | \( 1 - 3.69iT - 37T^{2} \) |
| 41 | \( 1 + 4.99T + 41T^{2} \) |
| 43 | \( 1 - 0.469iT - 43T^{2} \) |
| 47 | \( 1 - 1.69T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 9.39iT - 59T^{2} \) |
| 61 | \( 1 + 0.236iT - 61T^{2} \) |
| 67 | \( 1 - 5.42iT - 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 + 7.17T + 73T^{2} \) |
| 79 | \( 1 - 9.11T + 79T^{2} \) |
| 83 | \( 1 - 5.28iT - 83T^{2} \) |
| 89 | \( 1 - 8.28T + 89T^{2} \) |
| 97 | \( 1 - 0.681T + 97T^{2} \) |
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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
−8.918718067320164963112543950545, −8.218964459707561097803877434458, −7.48821432990272963122426736710, −6.45394184351885328128765018358, −5.77819783686184986492176632332, −5.05766033704367076649312770175, −4.15615933866627202902084891382, −3.45671018499971404588368212926, −2.48389481088392616039839896671, −1.13859818177621059664195502589,
0.34020528299563679366481005992, 1.78470761544770175316788623526, 2.69020150987569993868040900373, 3.46864775551418446172987815755, 4.66393928434396498604484177691, 5.28186746350675258151732258594, 6.48635927808158982913422966966, 6.93646450921596649324510745445, 7.28938251798930415309173575486, 8.517483124844360033110454517634
