| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 35-s + 37-s + 41-s − 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 57-s − 59-s + 61-s − 63-s − 67-s + ⋯ |
| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s + 15-s − 17-s + 19-s − 21-s − 23-s + 25-s + 27-s + 29-s + 31-s − 35-s + 37-s + 41-s − 43-s + 45-s + 47-s + 49-s − 51-s − 53-s + 57-s − 59-s + 61-s − 63-s − 67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1144 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.499708487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.499708487\) |
| \(L(1)\) |
\(\approx\) |
\(1.647645948\) |
| \(L(1)\) |
\(\approx\) |
\(1.647645948\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.33775353107266660656646488284, −20.36051065492380229326047032402, −19.91015960548435914033785769945, −19.09105002143157260549876443667, −18.22559006089300500543866108038, −17.65617778971382614813689919312, −16.49244967246512892092984549398, −15.842171640054803238181663216897, −15.105465685931672275630477626303, −13.88625759123119247466770737208, −13.78887841385238755483358953292, −12.886411853055073232270615852944, −12.14672742245607013241280386304, −10.785955483894877849979543964611, −9.85660317265552999254082707493, −9.53026136067926167463579126332, −8.68022778694939209452590374112, −7.73979467051055968602324524005, −6.67576949047891732393373086813, −6.155562207894656451151604512933, −4.90692343936839733699367597150, −3.9120074812077987019771254653, −2.860835426677091520307598979971, −2.32705750349022992095069986121, −1.11520769973581208491720193790,
1.11520769973581208491720193790, 2.32705750349022992095069986121, 2.860835426677091520307598979971, 3.9120074812077987019771254653, 4.90692343936839733699367597150, 6.155562207894656451151604512933, 6.67576949047891732393373086813, 7.73979467051055968602324524005, 8.68022778694939209452590374112, 9.53026136067926167463579126332, 9.85660317265552999254082707493, 10.785955483894877849979543964611, 12.14672742245607013241280386304, 12.886411853055073232270615852944, 13.78887841385238755483358953292, 13.88625759123119247466770737208, 15.105465685931672275630477626303, 15.842171640054803238181663216897, 16.49244967246512892092984549398, 17.65617778971382614813689919312, 18.22559006089300500543866108038, 19.09105002143157260549876443667, 19.91015960548435914033785769945, 20.36051065492380229326047032402, 21.33775353107266660656646488284
