L(s) = 1 | + (−0.986 − 0.164i)2-s + (0.546 + 0.837i)3-s + (0.945 + 0.324i)4-s + (0.789 + 0.614i)5-s + (−0.401 − 0.915i)6-s + (0.945 − 0.324i)7-s + (−0.879 − 0.475i)8-s + (−0.401 + 0.915i)9-s + (−0.677 − 0.735i)10-s + (−0.401 − 0.915i)11-s + (0.245 + 0.969i)12-s + (0.546 + 0.837i)13-s + (−0.986 + 0.164i)14-s + (−0.0825 + 0.996i)15-s + (0.789 + 0.614i)16-s + (0.945 − 0.324i)17-s + ⋯ |
L(s) = 1 | + (−0.986 − 0.164i)2-s + (0.546 + 0.837i)3-s + (0.945 + 0.324i)4-s + (0.789 + 0.614i)5-s + (−0.401 − 0.915i)6-s + (0.945 − 0.324i)7-s + (−0.879 − 0.475i)8-s + (−0.401 + 0.915i)9-s + (−0.677 − 0.735i)10-s + (−0.401 − 0.915i)11-s + (0.245 + 0.969i)12-s + (0.546 + 0.837i)13-s + (−0.986 + 0.164i)14-s + (−0.0825 + 0.996i)15-s + (0.789 + 0.614i)16-s + (0.945 − 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.589 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.161183892 + 0.5897447171i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.161183892 + 0.5897447171i\) |
\(L(1)\) |
\(\approx\) |
\(1.014689172 + 0.2979471320i\) |
\(L(1)\) |
\(\approx\) |
\(1.014689172 + 0.2979471320i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 + (-0.986 - 0.164i)T \) |
| 3 | \( 1 + (0.546 + 0.837i)T \) |
| 5 | \( 1 + (0.789 + 0.614i)T \) |
| 7 | \( 1 + (0.945 - 0.324i)T \) |
| 11 | \( 1 + (-0.401 - 0.915i)T \) |
| 13 | \( 1 + (0.546 + 0.837i)T \) |
| 17 | \( 1 + (0.945 - 0.324i)T \) |
| 23 | \( 1 + (0.546 - 0.837i)T \) |
| 29 | \( 1 + (0.945 - 0.324i)T \) |
| 31 | \( 1 + (-0.986 + 0.164i)T \) |
| 37 | \( 1 + (-0.401 - 0.915i)T \) |
| 41 | \( 1 + (-0.0825 + 0.996i)T \) |
| 43 | \( 1 + (-0.677 + 0.735i)T \) |
| 47 | \( 1 + (-0.401 - 0.915i)T \) |
| 53 | \( 1 + (-0.401 - 0.915i)T \) |
| 59 | \( 1 + (-0.0825 + 0.996i)T \) |
| 61 | \( 1 + (-0.879 + 0.475i)T \) |
| 67 | \( 1 + (-0.879 - 0.475i)T \) |
| 71 | \( 1 + (-0.879 - 0.475i)T \) |
| 73 | \( 1 + (0.945 - 0.324i)T \) |
| 79 | \( 1 + (-0.677 + 0.735i)T \) |
| 83 | \( 1 + (0.789 - 0.614i)T \) |
| 89 | \( 1 + (0.945 + 0.324i)T \) |
| 97 | \( 1 + (-0.879 + 0.475i)T \) |
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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
−24.96630517479532215257614924162, −23.94919816661019715692899930109, −23.3714913562909891931919573937, −21.54859382598143880620527323539, −20.51830587962932421911526361581, −20.43927014994633057209411660837, −19.09898542265656607967885154151, −18.224263501091024430297130029066, −17.66324892428119959657720924346, −17.055405603381293999536490459851, −15.6168370017178703178775125979, −14.836146752416721656143969046177, −13.879030801256999209117136093863, −12.70667178646013762763717203936, −12.0274480031715825535460007125, −10.71468578321186215369616402743, −9.70705272508191214263868315196, −8.760862296138558836204733453412, −8.062932403134732378889883031262, −7.26113615552590868645181622128, −5.97828375389778703295211350471, −5.15909314017882491065718273799, −3.04191729461333027024111729929, −1.84028373203087460057576163089, −1.2488078413361838980330787160,
1.488113584429362571220148257859, 2.62554006637230211269372323225, 3.56740408103303743925724031047, 5.096524619542517388796038662566, 6.29447866770370619941134947160, 7.54477723873054772357234835157, 8.47297466131622134155550622739, 9.24095391916905683620672592818, 10.30783780801548765146845587533, 10.83068971884027166809245499673, 11.64525584095349931286555045727, 13.41075204611254452483501731511, 14.30582978184892843013762435016, 14.95922800983909388572938615642, 16.32881858804631787324334734352, 16.684059137672857787198518803016, 17.93459731705171347794423649305, 18.59913668909654052098862548842, 19.4912493148609314405742171268, 20.63990320613687951140534097944, 21.33930029928211835079354862138, 21.51935033332077898947491211508, 23.06444948484377201393932816271, 24.34219252096523880900843320154, 25.17197152822837861973706000728