| L(s) = 1 | − 2.82i·2-s − 8.00·4-s − 83·7-s + 22.6i·8-s − 80.6i·11-s − 41·13-s + 234. i·14-s + 64.0·16-s + 513. i·17-s − 139·19-s − 228·22-s − 224. i·23-s + 115. i·26-s + 664.·28-s − 674. i·29-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.69·7-s + 0.353i·8-s − 0.666i·11-s − 0.242·13-s + 1.19i·14-s + 0.250·16-s + 1.77i·17-s − 0.385·19-s − 0.471·22-s − 0.425i·23-s + 0.171i·26-s + 0.846·28-s − 0.802i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.076426725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076426725\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 83T + 2.40e3T^{2} \) |
| 11 | \( 1 + 80.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 41T + 2.85e4T^{2} \) |
| 17 | \( 1 - 513. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 139T + 1.30e5T^{2} \) |
| 23 | \( 1 + 224. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 674. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.05e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.67e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 831. iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 2.51e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 2.94e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 390. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 750. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.82e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 7.25e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 6.90e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 9.29e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.98e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 6.07e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.79e3T + 8.85e7T^{2} \) |
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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
−10.44046630923034849601665517538, −9.586548407013225300273707336271, −8.857858904524668079703929288711, −7.77999410508680395803883763916, −6.39496174273941335776903917652, −5.83644049865966128632431288171, −4.17007130197752885590556317704, −3.39008991785455277909617283627, −2.27698905395963172246267944663, −0.61683659236779295392200735955,
0.51653426168262493240329966393, 2.59181272150738633398951179649, 3.71982074317332032159628536422, 4.95935506548344604362020961435, 5.99295224600861205571582951398, 6.97709441275613173985812448423, 7.44434584397889287308884776917, 8.964516087520742296047036227631, 9.515893346906762112575587115335, 10.21785883527478271901452652081
