L(s) = 1 | − 4-s + 16-s + 2·17-s + 2·29-s + 2·37-s + 2·41-s − 2·61-s − 64-s − 2·68-s − 2·73-s − 81-s + 2·97-s − 2·109-s + 2·113-s − 2·116-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯ |
L(s) = 1 | − 4-s + 16-s + 2·17-s + 2·29-s + 2·37-s + 2·41-s − 2·61-s − 64-s − 2·68-s − 2·73-s − 81-s + 2·97-s − 2·109-s + 2·113-s − 2·116-s + 127-s + 131-s + 137-s + 139-s − 2·148-s + 149-s + 151-s + 157-s + 163-s − 2·164-s + 167-s − 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2890000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.058398888\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.058398888\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.621693204528338566444015964958, −9.551540258418744755566180179705, −8.745725562750514140642658078143, −8.739553007836925901026062919836, −8.049062540152930126437150533153, −7.904385708032775234005493040217, −7.29929746361582017316779981568, −7.29311814974168229559373964873, −6.22981557822582367690271098918, −5.97799437245419381017560834066, −5.90844963570508377150311121060, −5.13609979479622221374425467642, −4.72839447478391209968383154611, −4.42529315141504950159012268127, −3.94661804210477456945265977563, −3.32032866383993487822211509574, −2.93263388386914010013786930941, −2.46015059581200257226223753034, −1.28997041010415709024147484752, −0.987411808212863579655714939590,
0.987411808212863579655714939590, 1.28997041010415709024147484752, 2.46015059581200257226223753034, 2.93263388386914010013786930941, 3.32032866383993487822211509574, 3.94661804210477456945265977563, 4.42529315141504950159012268127, 4.72839447478391209968383154611, 5.13609979479622221374425467642, 5.90844963570508377150311121060, 5.97799437245419381017560834066, 6.22981557822582367690271098918, 7.29311814974168229559373964873, 7.29929746361582017316779981568, 7.904385708032775234005493040217, 8.049062540152930126437150533153, 8.739553007836925901026062919836, 8.745725562750514140642658078143, 9.551540258418744755566180179705, 9.621693204528338566444015964958