L(s) = 1 | + (−0.707 + 0.707i)3-s − 7-s − i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − 23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s − i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s − 7-s − i·9-s + (−0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − i·17-s + (−0.707 − 0.707i)19-s + (0.707 − 0.707i)21-s − 23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s − 31-s − i·33-s + (−0.707 − 0.707i)37-s − i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01434672179 + 0.07946721892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01434672179 + 0.07946721892i\) |
\(L(1)\) |
\(\approx\) |
\(0.5014135764 + 0.1289956140i\) |
\(L(1)\) |
\(\approx\) |
\(0.5014135764 + 0.1289956140i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + (-0.707 + 0.707i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + (-0.707 - 0.707i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.707 - 0.707i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (-0.707 - 0.707i)T \) |
| 41 | \( 1 + iT \) |
| 43 | \( 1 + (0.707 + 0.707i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.707 - 0.707i)T \) |
| 59 | \( 1 + (-0.707 + 0.707i)T \) |
| 61 | \( 1 + (0.707 + 0.707i)T \) |
| 67 | \( 1 + (0.707 - 0.707i)T \) |
| 71 | \( 1 + iT \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + (0.707 - 0.707i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + iT \) |
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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
−27.47154262855507806149596151162, −26.15393923141976460143802814856, −25.32685432989472070920025925557, −24.1611641306812201323104026169, −23.50418879580785515265623559366, −22.386656611695262754941493692180, −21.78039410125735078934945152253, −20.20762520010357395382381142367, −19.16551957014875198121345168586, −18.51958249350204016684345916859, −17.27764743248213024606588745724, −16.50669449583835489936493332916, −15.45633446569099604976603066406, −13.979992082213404143461251720113, −12.807407314757844309011800198066, −12.40303200141454885105371420132, −10.85549560130987711471079101280, −10.12284789960212253587844724852, −8.450876832163183792146529506562, −7.37616670755221132499709783595, −6.16598330497692058024087981875, −5.39680002967222887571845194176, −3.577243813047840912328796192228, −2.06577950341454943502600405157, −0.06768012176402300070191595438,
2.50518602063757181244917705505, 4.028856621838587668814407844805, 5.08328056982625538899133901446, 6.326651494816406864694177100602, 7.37701641321528120128902462558, 9.27952198158727838087543058656, 9.838666395608220720590379381401, 11.008736904093074279398367787, 12.13494675736584108175121744426, 13.04734825024154412931685622188, 14.52471741608865473592954381627, 15.66334233573805815243740378791, 16.30967889735451683733424441620, 17.35322396707411441554908286810, 18.36052215616774458762247013461, 19.58774954107345883350483116789, 20.64023080795777901877468725426, 21.67740705465832644605225980619, 22.51499913092603884451430381762, 23.2747794101190504299326375357, 24.303577434440594904215867180261, 25.818846988146875957874074930932, 26.32462013298385202829092108049, 27.48288093400341670954479876371, 28.44142693178944696109685343942