L(s) = 1 | − 5.40i·2-s − 13.2·4-s + 5.33i·5-s − 40.2·7-s − 14.9i·8-s + 28.8·10-s + 236. i·11-s + 228.·13-s + 217. i·14-s − 292.·16-s − 143. i·17-s − 422.·19-s − 70.6i·20-s + 1.27e3·22-s − 486. i·23-s + ⋯ |
L(s) = 1 | − 1.35i·2-s − 0.827·4-s + 0.213i·5-s − 0.821·7-s − 0.233i·8-s + 0.288·10-s + 1.95i·11-s + 1.35·13-s + 1.11i·14-s − 1.14·16-s − 0.495i·17-s − 1.17·19-s − 0.176i·20-s + 2.63·22-s − 0.918i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.622864848\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.622864848\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
good | 2 | \( 1 + 5.40iT - 16T^{2} \) |
| 5 | \( 1 - 5.33iT - 625T^{2} \) |
| 7 | \( 1 + 40.2T + 2.40e3T^{2} \) |
| 11 | \( 1 - 236. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 228.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 143. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 422.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 486. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 464. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.10e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 60.3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 404. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.54e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.12e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 5.19e3iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 126.T + 1.38e7T^{2} \) |
| 67 | \( 1 - 5.06e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 3.20e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.56e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 71.8T + 3.89e7T^{2} \) |
| 83 | \( 1 + 4.14e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 7.28e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.10e4T + 8.85e7T^{2} \) |
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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
−10.08942970161495216195106131487, −9.399851924673973465017684643058, −8.401770140698098257480315855139, −6.91765642980037655688903852131, −6.43576127913233089521317464443, −4.68975764747875425576770414142, −3.89333136064062724777559323089, −2.75476029013549012709330001690, −1.88056704995904762259306376091, −0.50546674178132231193833836464,
0.975917042975538816801649377351, 2.96198360199834956789910908065, 4.03786961190387217237117834893, 5.47694137565113610462087542794, 6.19348017901942187693462680329, 6.63321838913569255941471668952, 8.051284724222524078000748617689, 8.550990754438057334653423282909, 9.243267393461456799796284347202, 10.76004027017702725068471282879