| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 22-s − 23-s + 24-s + 26-s + 27-s − 28-s + 29-s + 31-s + 32-s − 33-s + 34-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s − 11-s + 12-s + 13-s − 14-s + 16-s + 17-s + 18-s − 19-s − 21-s − 22-s − 23-s + 24-s + 26-s + 27-s − 28-s + 29-s + 31-s + 32-s − 33-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(7.449564464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.449564464\) |
| \(L(1)\) |
\(\approx\) |
\(2.770316498\) |
| \(L(1)\) |
\(\approx\) |
\(2.770316498\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 643 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.86342779998923272054752067521, −18.44500246777093104445203221917, −17.148156053956261408824080676653, −16.25314048620783295372729797429, −15.72115292936187578108138688401, −15.387828875802763031542423080420, −14.39796217963025177073580932087, −13.78825372860695615494494091663, −13.30886212644369591831977994986, −12.58818147679978652651534818527, −12.15915748559184923068676347829, −10.90410199316932537945350943244, −10.269382807565392861906795271256, −9.74313534462776086897755016350, −8.56992691124748734616782562969, −7.99127432332524565487274184461, −7.25873643796003191951989758411, −6.28323225154295500981627604008, −5.913919988888241190244185868180, −4.6830298961824691368662257501, −4.03842572117548901557513631479, −3.21013889655314043543029759314, −2.765317950743839927095535983344, −1.89499207178163743146015363360, −0.80834638004106956861801105373,
0.80834638004106956861801105373, 1.89499207178163743146015363360, 2.765317950743839927095535983344, 3.21013889655314043543029759314, 4.03842572117548901557513631479, 4.6830298961824691368662257501, 5.913919988888241190244185868180, 6.28323225154295500981627604008, 7.25873643796003191951989758411, 7.99127432332524565487274184461, 8.56992691124748734616782562969, 9.74313534462776086897755016350, 10.269382807565392861906795271256, 10.90410199316932537945350943244, 12.15915748559184923068676347829, 12.58818147679978652651534818527, 13.30886212644369591831977994986, 13.78825372860695615494494091663, 14.39796217963025177073580932087, 15.387828875802763031542423080420, 15.72115292936187578108138688401, 16.25314048620783295372729797429, 17.148156053956261408824080676653, 18.44500246777093104445203221917, 18.86342779998923272054752067521
