L(s) = 1 | + (−0.0665 − 0.997i)2-s + (−0.104 − 0.994i)3-s + (−0.991 + 0.132i)4-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (0.235 + 0.971i)12-s + (0.564 − 0.825i)13-s + (0.964 − 0.263i)16-s + (−0.797 + 0.603i)17-s + (0.272 + 0.962i)18-s + (−0.290 − 0.956i)19-s + (−0.0475 + 0.998i)23-s + (0.953 − 0.299i)24-s + (−0.861 − 0.508i)26-s + (0.309 + 0.951i)27-s + ⋯ |
L(s) = 1 | + (−0.0665 − 0.997i)2-s + (−0.104 − 0.994i)3-s + (−0.991 + 0.132i)4-s + (−0.985 + 0.170i)6-s + (0.198 + 0.980i)8-s + (−0.978 + 0.207i)9-s + (0.235 + 0.971i)12-s + (0.564 − 0.825i)13-s + (0.964 − 0.263i)16-s + (−0.797 + 0.603i)17-s + (0.272 + 0.962i)18-s + (−0.290 − 0.956i)19-s + (−0.0475 + 0.998i)23-s + (0.953 − 0.299i)24-s + (−0.861 − 0.508i)26-s + (0.309 + 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2830247561 - 0.2203112614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2830247561 - 0.2203112614i\) |
\(L(1)\) |
\(\approx\) |
\(0.4832614572 - 0.5682226606i\) |
\(L(1)\) |
\(\approx\) |
\(0.4832614572 - 0.5682226606i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.0665 - 0.997i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 13 | \( 1 + (0.564 - 0.825i)T \) |
| 17 | \( 1 + (-0.797 + 0.603i)T \) |
| 19 | \( 1 + (-0.290 - 0.956i)T \) |
| 23 | \( 1 + (-0.0475 + 0.998i)T \) |
| 29 | \( 1 + (-0.0855 - 0.996i)T \) |
| 31 | \( 1 + (0.851 + 0.524i)T \) |
| 37 | \( 1 + (0.179 + 0.983i)T \) |
| 41 | \( 1 + (-0.466 - 0.884i)T \) |
| 43 | \( 1 + (0.415 - 0.909i)T \) |
| 47 | \( 1 + (0.272 - 0.962i)T \) |
| 53 | \( 1 + (-0.964 - 0.263i)T \) |
| 59 | \( 1 + (0.532 + 0.846i)T \) |
| 61 | \( 1 + (-0.830 - 0.556i)T \) |
| 67 | \( 1 + (0.786 + 0.618i)T \) |
| 71 | \( 1 + (0.516 - 0.856i)T \) |
| 73 | \( 1 + (-0.988 - 0.151i)T \) |
| 79 | \( 1 + (0.595 + 0.803i)T \) |
| 83 | \( 1 + (-0.774 - 0.633i)T \) |
| 89 | \( 1 + (-0.981 - 0.189i)T \) |
| 97 | \( 1 + (-0.736 - 0.676i)T \) |
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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
−18.67959235917030347380904489512, −18.108383482551162968872470727184, −17.32843652091170443266697462111, −16.69223580801311452921863621316, −16.096922128420097173159064022611, −15.79734282120518585966218340420, −14.83248123135557825947953983863, −14.38444976898151764375630230486, −13.82505771268854484058732762005, −12.92599229282442798862952366374, −12.137086004806593953774934274444, −11.15111139209674209396509345844, −10.64554765363613773879039481630, −9.67005514875503209518879253057, −9.29441315275609942654363431468, −8.50414040675597867953736628403, −7.993566444609318480878208441742, −6.865118629024692189640058429803, −6.293937359476015662553023391000, −5.65982933613564226838988419839, −4.65778292542546872162040949475, −4.356555283864264872440631597149, −3.54077571279841016830172634285, −2.5476735307382743505684710168, −1.22066003704643958325706951716,
0.12031874519195723813169941710, 1.05074651198619724485057492281, 1.84906204846364636554029864447, 2.58598789089143840207342315198, 3.30852272322598299921367677687, 4.18342679197260789760107111752, 5.13261649094365564182046915811, 5.81985922198675827727778610047, 6.64243490491395050285595896239, 7.504539652121429856618955852652, 8.37558837308771835560454100948, 8.65857127393737162424057744437, 9.638334731944180207650990875993, 10.50732344413934926763433006121, 11.07992156285376903928529494150, 11.716329743387982988780214140569, 12.34071635723678699643149238387, 13.141182579465357193506947435767, 13.46526487988828939356754223829, 14.0202678863828571674989588963, 15.14408811176754299601163274501, 15.61794471939214054668133451414, 17.02509230663741481112794812179, 17.37241608266188980376658085892, 17.90408360592154393822485729803