| L(s) = 1 | − 4·2-s + 9·3-s + 16·4-s − 72·5-s − 36·6-s + 49·7-s − 64·8-s + 81·9-s + 288·10-s − 414·11-s + 144·12-s − 1.05e3·13-s − 196·14-s − 648·15-s + 256·16-s − 1.84e3·17-s − 324·18-s + 236·19-s − 1.15e3·20-s + 441·21-s + 1.65e3·22-s + 2.89e3·23-s − 576·24-s + 2.05e3·25-s + 4.21e3·26-s + 729·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.28·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.910·10-s − 1.03·11-s + 0.288·12-s − 1.72·13-s − 0.267·14-s − 0.743·15-s + 1/4·16-s − 1.55·17-s − 0.235·18-s + 0.149·19-s − 0.643·20-s + 0.218·21-s + 0.729·22-s + 1.14·23-s − 0.204·24-s + 0.658·25-s + 1.22·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 3 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 72 T + p^{5} T^{2} \) |
| 11 | \( 1 + 414 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1054 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1848 T + p^{5} T^{2} \) |
| 19 | \( 1 - 236 T + p^{5} T^{2} \) |
| 23 | \( 1 - 126 p T + p^{5} T^{2} \) |
| 29 | \( 1 + 6522 T + p^{5} T^{2} \) |
| 31 | \( 1 - 200 p T + p^{5} T^{2} \) |
| 37 | \( 1 - 9650 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8484 T + p^{5} T^{2} \) |
| 43 | \( 1 + 10804 T + p^{5} T^{2} \) |
| 47 | \( 1 - 60 T + p^{5} T^{2} \) |
| 53 | \( 1 - 22506 T + p^{5} T^{2} \) |
| 59 | \( 1 + 28176 T + p^{5} T^{2} \) |
| 61 | \( 1 + 35194 T + p^{5} T^{2} \) |
| 67 | \( 1 + 28216 T + p^{5} T^{2} \) |
| 71 | \( 1 + 6642 T + p^{5} T^{2} \) |
| 73 | \( 1 + 52090 T + p^{5} T^{2} \) |
| 79 | \( 1 - 43340 T + p^{5} T^{2} \) |
| 83 | \( 1 - 25716 T + p^{5} T^{2} \) |
| 89 | \( 1 - 98724 T + p^{5} T^{2} \) |
| 97 | \( 1 + 148954 T + p^{5} T^{2} \) |
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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
−14.97144571870793356890188229748, −13.14237925431625492625491665031, −11.83827142985551238043436327553, −10.74422271212181268953492435797, −9.261427045994095972582404016536, −7.967314936873959005499244248623, −7.25029656661878260973173379245, −4.62980474777732664962063367475, −2.61270403452721611500665532978, 0,
2.61270403452721611500665532978, 4.62980474777732664962063367475, 7.25029656661878260973173379245, 7.967314936873959005499244248623, 9.261427045994095972582404016536, 10.74422271212181268953492435797, 11.83827142985551238043436327553, 13.14237925431625492625491665031, 14.97144571870793356890188229748
