L(s) = 1 | + 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s − 2·80-s − 4·95-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 2·5-s − 16-s − 2·19-s + 3·25-s + 2·49-s − 2·80-s − 4·95-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.257104105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.257104105\) |
\(L(1)\) |
|
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
−10.45792011474811071120776346473, −10.45086148821719386969462945688, −9.623923462398388958674827630161, −9.406427590787638692703192745680, −8.855992603957737578040280210684, −8.770249463260139601258757786070, −8.234173153575635250964562267969, −7.53874129702747483862315240467, −7.01667909629407094397518069355, −6.54735645376233508662178637077, −6.33609500404421226072248267111, −5.88672294809600988220141059621, −5.31776636594026645937298083103, −4.98490716495764179214499644337, −4.31432945832676453282242500406, −3.93930107019701489209123059048, −2.94017852514384129976399413879, −2.35713804217991572148981181520, −2.13367658955366694371033754489, −1.30458727967014322993710946472,
1.30458727967014322993710946472, 2.13367658955366694371033754489, 2.35713804217991572148981181520, 2.94017852514384129976399413879, 3.93930107019701489209123059048, 4.31432945832676453282242500406, 4.98490716495764179214499644337, 5.31776636594026645937298083103, 5.88672294809600988220141059621, 6.33609500404421226072248267111, 6.54735645376233508662178637077, 7.01667909629407094397518069355, 7.53874129702747483862315240467, 8.234173153575635250964562267969, 8.770249463260139601258757786070, 8.855992603957737578040280210684, 9.406427590787638692703192745680, 9.623923462398388958674827630161, 10.45086148821719386969462945688, 10.45792011474811071120776346473