L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + 21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − 31-s + (0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + ⋯ |
L(s) = 1 | + (0.866 − 0.5i)3-s + (0.866 + 0.5i)7-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)19-s + 21-s + (0.866 − 0.5i)23-s − i·27-s + (0.5 + 0.866i)29-s − 31-s + (0.866 + 0.5i)33-s + (0.866 − 0.5i)37-s + (−0.5 − 0.866i)41-s + (−0.866 − 0.5i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.765813706 - 0.2113404016i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765813706 - 0.2113404016i\) |
\(L(1)\) |
\(\approx\) |
\(1.476215425 - 0.1374062835i\) |
\(L(1)\) |
\(\approx\) |
\(1.476215425 - 0.1374062835i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.866 + 0.5i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.866 - 0.5i)T \) |
| 19 | \( 1 + (-0.5 + 0.866i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.866 - 0.5i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.866 - 0.5i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−26.05947455871633554169791926348, −24.98476679497765309193187512915, −24.268005149357403939516108462260, −23.32900345077443345986347217085, −21.789992568523660838668095610176, −21.526528602121779709078332345184, −20.31757428670264657533713402195, −19.69667252692078543807921311262, −18.73866676479374886609010420613, −17.45973357422259110956842546431, −16.65348561175099027169762563810, −15.44215135592472121383385155626, −14.74016729261667038352721932385, −13.77484696709948139517914806955, −13.08523384580037628957232980079, −11.339567253735096825713911672512, −10.80614494712094477442704092029, −9.47010634339518865281956646529, −8.596003370966027509426737883581, −7.7684849072117759194584494779, −6.49426592526206037940687675618, −4.90560493889550378368312030095, −4.07237614745304444014497652748, −2.8688927440393070488999935866, −1.500464515222513451833054181,
1.56770702719305979460957515404, 2.43443273398146362921827549267, 3.89374106234390505390553479083, 5.03663906614722224575279915663, 6.58056107745934306520086870577, 7.46342147368327215835006319815, 8.58781886454260775354200544157, 9.21974915111300868477207812514, 10.57964475004329991754727035949, 11.87267765290051827378669956631, 12.62486657504094859814402170828, 13.72608729278680841820381798352, 14.78340227892520767021431925250, 15.11035922426178830840933947789, 16.60880951515346421340657472624, 17.88271120826290991508972611483, 18.35401006214829343301274268908, 19.501069042742028809297627188459, 20.34898726618220501318564096336, 21.051505839833761434760628204219, 22.13099652225185990206958922408, 23.30298787555012503219915896329, 24.21363660695672242423877683054, 25.10426857015254291395344355752, 25.498700109251755867468120240404