L(s) = 1 | + 3.46i·5-s − 7-s + 3.31i·11-s − 3.10i·13-s + 1.40·17-s + 4.80i·19-s + 8.79·23-s − 6.99·25-s − 9.87i·29-s + 7.83·31-s − 3.46i·35-s − 5.42i·37-s + 11.5·41-s + 7.03i·43-s + 11.2·47-s + ⋯ |
L(s) = 1 | + 1.54i·5-s − 0.377·7-s + 1.00i·11-s − 0.862i·13-s + 0.340·17-s + 1.10i·19-s + 1.83·23-s − 1.39·25-s − 1.83i·29-s + 1.40·31-s − 0.585i·35-s − 0.891i·37-s + 1.80·41-s + 1.07i·43-s + 1.64·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0841 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.010118439\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.010118439\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 11 | \( 1 - 3.31iT - 11T^{2} \) |
| 13 | \( 1 + 3.10iT - 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 4.80iT - 19T^{2} \) |
| 23 | \( 1 - 8.79T + 23T^{2} \) |
| 29 | \( 1 + 9.87iT - 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 + 5.42iT - 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 7.03iT - 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6.51iT - 53T^{2} \) |
| 59 | \( 1 - 3.89iT - 59T^{2} \) |
| 61 | \( 1 - 9.42iT - 61T^{2} \) |
| 67 | \( 1 + 0.909iT - 67T^{2} \) |
| 71 | \( 1 + 6.42T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 0.370iT - 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 - 1.63T + 97T^{2} \) |
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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
−7.892801525939083640281108753508, −7.56476854204422114803600451715, −6.91778622986237215403568651780, −6.11826264243551157333896108286, −5.68894081372214641686686189158, −4.48469993099034641930299915851, −3.79665341468591063869148499350, −2.72770535197689784284254344836, −2.57202557318344112564290223133, −1.01400634401365573685976491479,
0.66795814605757130207453666950, 1.24630235135724015959634406043, 2.59963805071479922986838877274, 3.43880284343233088925463534309, 4.40690065712184958411582342805, 5.01436719459181746087412059287, 5.54988613317088292615206031650, 6.54469344614984474672898993762, 7.08366484661880156994782609980, 8.089252632334886377867985666711