L(s) = 1 | + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 − 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (0.841 + 0.540i)21-s + ⋯ |
L(s) = 1 | + (0.985 + 0.170i)2-s + (0.809 − 0.587i)3-s + (0.941 + 0.336i)4-s + (0.897 − 0.441i)6-s + (0.362 + 0.931i)7-s + (0.870 + 0.491i)8-s + (0.309 − 0.951i)9-s + (0.959 − 0.281i)12-s + (0.0285 − 0.999i)13-s + (0.198 + 0.980i)14-s + (0.774 + 0.633i)16-s + (−0.516 + 0.856i)17-s + (0.466 − 0.884i)18-s + (0.0855 − 0.996i)19-s + (0.841 + 0.540i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.613221411 - 0.05629184215i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.613221411 - 0.05629184215i\) |
\(L(1)\) |
\(\approx\) |
\(2.489627666 + 0.02580965937i\) |
\(L(1)\) |
\(\approx\) |
\(2.489627666 + 0.02580965937i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.985 + 0.170i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 7 | \( 1 + (0.362 + 0.931i)T \) |
| 13 | \( 1 + (0.0285 - 0.999i)T \) |
| 17 | \( 1 + (-0.516 + 0.856i)T \) |
| 19 | \( 1 + (0.0855 - 0.996i)T \) |
| 23 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.974 - 0.226i)T \) |
| 31 | \( 1 + (-0.564 + 0.825i)T \) |
| 37 | \( 1 + (-0.610 - 0.791i)T \) |
| 41 | \( 1 + (0.696 + 0.717i)T \) |
| 43 | \( 1 + (-0.415 - 0.909i)T \) |
| 47 | \( 1 + (0.466 + 0.884i)T \) |
| 53 | \( 1 + (-0.774 + 0.633i)T \) |
| 59 | \( 1 + (0.696 - 0.717i)T \) |
| 61 | \( 1 + (-0.985 + 0.170i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
| 71 | \( 1 + (-0.921 - 0.389i)T \) |
| 73 | \( 1 + (0.998 + 0.0570i)T \) |
| 79 | \( 1 + (-0.736 - 0.676i)T \) |
| 83 | \( 1 + (0.254 - 0.967i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (-0.993 + 0.113i)T \) |
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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
−22.924178046628590652018048734, −22.22543302099812424221195959763, −21.28999432274352716603143967832, −20.69650420540572425057065202609, −20.09899072260384799247794962957, −19.33299929661455468244677995599, −18.306607395265395487074397815122, −16.76590387848252864562018653669, −16.28276395581974236246418234691, −15.3879158260746763291402862122, −14.30395337449786651855560281635, −14.06601224675666331261772031975, −13.272319145017841510091828247368, −12.09053032530888992983614599088, −11.17143892559673981745714667055, −10.344759183538489244893333659060, −9.57693783069685314391491228207, −8.28755940707650832937355660009, −7.37925452287946237589777570415, −6.484640096761383098852751593679, −5.100553102679835722964718566792, −4.29520668697338781281222121061, −3.70940276527797456975086112678, −2.51365272563088554991474380810, −1.55753004524066391073552064869,
1.57629808705632793697180311386, 2.51007732092703416702596762519, 3.28687526115483110436928774889, 4.441331098507106861124571550724, 5.57732560247077943016424410568, 6.364713589875868196920421971312, 7.42938956173539112758187896284, 8.22429738699935874390556484585, 9.01263853105446548304582684842, 10.41578282716234167377446229596, 11.48186746718173205766577093222, 12.4148346225526891118493151262, 12.90525495439569815090149222926, 13.861438913448359814296015103, 14.58602154094281278956918239193, 15.460963356282280538433718985707, 15.7905518105160374066219880589, 17.49954521864785915571825331406, 17.93183140677732382611152915054, 19.25460829791732151094432938319, 19.84643269365514592654772825518, 20.63410361157414149326221764851, 21.59553969921488392881158352061, 22.047070153449651759308596191129, 23.264780976668927544448367556960