L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (0.866 + 0.5i)19-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + i·5-s + (0.866 + 0.5i)6-s + (−0.866 − 0.5i)7-s + i·8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s − 12-s + 14-s + (0.866 − 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − i·18-s + (0.866 + 0.5i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2889488098 + 0.2964985948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2889488098 + 0.2964985948i\) |
\(L(1)\) |
\(\approx\) |
\(0.5008886454 + 0.1268363814i\) |
\(L(1)\) |
\(\approx\) |
\(0.5008886454 + 0.1268363814i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + iT \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (-0.866 - 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (-0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
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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
−28.2511403817126314934554038385, −27.20005927321229592026808855621, −26.39427653045273366725477321300, −25.30681654409985872683810112721, −24.34758567984641413512488190249, −22.72186817046662120523934182082, −21.97164907503055197303491498796, −20.83217691958936888527391006876, −20.25829328982433202898403950374, −19.09836151573842953095471254039, −17.8611513421152623735078586940, −16.908504023059659189786282705272, −16.07545197254423839538757772047, −15.48413118878754345473421450230, −13.36456184217038050491918396918, −12.21466660451499753153601929018, −11.497143651629879380626935774348, −10.11074717478164107611675019357, −9.32004423720021525683676425631, −8.59545641129922321305767347052, −6.84544615220317604662355899820, −5.41640807540703984081121232744, −4.065673427411477802622228083811, −2.66927412301754200105765668114, −0.524098281670507298995813375147,
1.51752741336828903508777699201, 3.12988208258577825893234458674, 5.503617661483251237598407065901, 6.67796865105198330824618797838, 7.09572891366971251026043789084, 8.36350954639182555041902002879, 9.95358167301379851124959033173, 10.74471010814774866571088257689, 11.82884111816652185518463980596, 13.327100541014025314977608925438, 14.32102045615135898002913103914, 15.5991926569412771844840691744, 16.65648484452825800790319185435, 17.62052779329595995208755878944, 18.41110987208063597769578178493, 19.322261687571690865944540756807, 19.94057961407214203307715080275, 21.89379158221126870381589311390, 23.02666650547530976957401196597, 23.54937989377774721816283286859, 24.80311655978768964533288997984, 25.67108473988350187257424629854, 26.44671720986417811049385584810, 27.44652866755151293382522273102, 28.74994189664554799944157858502