| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s − 2·9-s − 12-s + 4·13-s − 14-s + 16-s − 3·17-s − 2·18-s + 2·19-s + 21-s − 24-s + 4·26-s + 5·27-s − 28-s − 6·29-s − 31-s + 32-s − 3·34-s − 2·36-s + 4·37-s + 2·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.471·18-s + 0.458·19-s + 0.218·21-s − 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.188·28-s − 1.11·29-s − 0.179·31-s + 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.657·37-s + 0.324·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155925522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155925522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 293 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−13.67481946359248, −13.19380920959247, −12.85545033758101, −12.30156717218092, −11.69316739767221, −11.30229784000668, −11.01777850719156, −10.53136913941066, −9.855872439585654, −9.193668062054901, −8.907365268025386, −8.062431951503150, −7.795436005378910, −6.890676111067594, −6.552747331698556, −6.031839775513853, −5.624016935473501, −5.102675529742911, −4.471712525909854, −3.849347253514660, −3.333157191238274, −2.772673899763548, −2.036897389109022, −1.283307741594124, −0.4301493974815623,
0.4301493974815623, 1.283307741594124, 2.036897389109022, 2.772673899763548, 3.333157191238274, 3.849347253514660, 4.471712525909854, 5.102675529742911, 5.624016935473501, 6.031839775513853, 6.552747331698556, 6.890676111067594, 7.795436005378910, 8.062431951503150, 8.907365268025386, 9.193668062054901, 9.855872439585654, 10.53136913941066, 11.01777850719156, 11.30229784000668, 11.69316739767221, 12.30156717218092, 12.85545033758101, 13.19380920959247, 13.67481946359248
