L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 43-s − 44-s + 46-s + 47-s + 49-s − 52-s + ⋯ |
L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 11-s − 13-s − 14-s + 16-s + 17-s + 19-s − 22-s + 23-s − 26-s − 28-s − 29-s + 31-s + 32-s + 34-s − 37-s + 38-s − 41-s − 43-s − 44-s + 46-s + 47-s + 49-s − 52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.847029343\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.847029343\) |
\(L(1)\) |
\(\approx\) |
\(1.622311470\) |
\(L(1)\) |
\(\approx\) |
\(1.622311470\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 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
−41.99740257490063314886123066149, −41.22823157698679018330044979235, −39.44089794397602895029589030001, −38.757341359970418830354799182704, −36.99584669791703414864511396216, −35.0970556892378917853228489627, −33.78117919887043491359601713057, −32.32338008181344539433639966495, −31.36470623994313228882952612958, −29.69227751983587214188771803184, −28.65701374833144389761432722966, −26.31976601519719112857065676519, −24.90350116904296767595196452334, −23.36799923867493537660362632137, −22.17639768821376108756020485300, −20.66236305141030677303888216364, −19.10349718525469465744305631936, −16.63355841618558638158823442508, −15.240408529039033166877888159850, −13.48938457315531804053322255533, −12.15969853381077081792324381745, −10.133748155228375797259200600348, −7.26763059826680725286077767720, −5.34319209367996891782367161036, −3.05701820980868915287415506904,
3.05701820980868915287415506904, 5.34319209367996891782367161036, 7.26763059826680725286077767720, 10.133748155228375797259200600348, 12.15969853381077081792324381745, 13.48938457315531804053322255533, 15.240408529039033166877888159850, 16.63355841618558638158823442508, 19.10349718525469465744305631936, 20.66236305141030677303888216364, 22.17639768821376108756020485300, 23.36799923867493537660362632137, 24.90350116904296767595196452334, 26.31976601519719112857065676519, 28.65701374833144389761432722966, 29.69227751983587214188771803184, 31.36470623994313228882952612958, 32.32338008181344539433639966495, 33.78117919887043491359601713057, 35.0970556892378917853228489627, 36.99584669791703414864511396216, 38.757341359970418830354799182704, 39.44089794397602895029589030001, 41.22823157698679018330044979235, 41.99740257490063314886123066149