| L(s) = 1 | + i·2-s − i·3-s − 4-s + 6-s − 0.732i·7-s − i·8-s − 9-s + 3.73·11-s + i·12-s − 2.73i·13-s + 0.732·14-s + 16-s + 6.19i·17-s − i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.276i·7-s − 0.353i·8-s − 0.333·9-s + 1.12·11-s + 0.288i·12-s − 0.757i·13-s + 0.195·14-s + 0.250·16-s + 1.50i·17-s − 0.235i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.784479443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.784479443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 0.732iT - 7T^{2} \) |
| 11 | \( 1 - 3.73T + 11T^{2} \) |
| 13 | \( 1 + 2.73iT - 13T^{2} \) |
| 17 | \( 1 - 6.19iT - 17T^{2} \) |
| 23 | \( 1 - 5.92iT - 23T^{2} \) |
| 29 | \( 1 - 1.73T + 29T^{2} \) |
| 31 | \( 1 + 2.46T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 2.92T + 41T^{2} \) |
| 43 | \( 1 + 8.19iT - 43T^{2} \) |
| 47 | \( 1 + 3.46iT - 47T^{2} \) |
| 53 | \( 1 + 1.73iT - 53T^{2} \) |
| 59 | \( 1 - 8.19T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 - 0.267iT - 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 + 2.46iT - 73T^{2} \) |
| 79 | \( 1 - 5.53T + 79T^{2} \) |
| 83 | \( 1 - 3.73iT - 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 15.1iT - 97T^{2} \) |
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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
−8.544854766610515869152510647630, −8.114938384923557941657690004186, −7.18318777424849916436107344966, −6.71122241124804970156069372035, −5.83406393139373014800194033839, −5.27875310449029642461354347961, −3.97931573118379837891004225629, −3.48233844102926234694035031009, −1.94022812470502664695990578927, −0.890269677368230293234200310857,
0.809795960787126667437717395981, 2.14941746668255282399398898964, 3.01974169640940254419882780082, 3.99477473172513420733561292919, 4.61296657343217588125430451159, 5.42151456533695700342534489877, 6.45465247125259621444879371186, 7.13569899733110462054180617564, 8.297143911405069362071908551079, 8.997312651915540537152279054835
