L(s) = 1 | + (0.796 + 0.605i)2-s + (0.468 + 0.883i)3-s + (0.267 + 0.963i)4-s + (−0.725 − 0.687i)5-s + (−0.161 + 0.986i)6-s + (0.647 − 0.762i)7-s + (−0.370 + 0.928i)8-s + (−0.561 + 0.827i)9-s + (−0.161 − 0.986i)10-s + (−0.856 − 0.515i)11-s + (−0.725 + 0.687i)12-s + (−0.561 − 0.827i)13-s + (0.976 − 0.214i)14-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (0.647 + 0.762i)17-s + ⋯ |
L(s) = 1 | + (0.796 + 0.605i)2-s + (0.468 + 0.883i)3-s + (0.267 + 0.963i)4-s + (−0.725 − 0.687i)5-s + (−0.161 + 0.986i)6-s + (0.647 − 0.762i)7-s + (−0.370 + 0.928i)8-s + (−0.561 + 0.827i)9-s + (−0.161 − 0.986i)10-s + (−0.856 − 0.515i)11-s + (−0.725 + 0.687i)12-s + (−0.561 − 0.827i)13-s + (0.976 − 0.214i)14-s + (0.267 − 0.963i)15-s + (−0.856 + 0.515i)16-s + (0.647 + 0.762i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.255 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.074766643 + 0.8280522620i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.074766643 + 0.8280522620i\) |
\(L(1)\) |
\(\approx\) |
\(1.300664010 + 0.6956238862i\) |
\(L(1)\) |
\(\approx\) |
\(1.300664010 + 0.6956238862i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.796 + 0.605i)T \) |
| 3 | \( 1 + (0.468 + 0.883i)T \) |
| 5 | \( 1 + (-0.725 - 0.687i)T \) |
| 7 | \( 1 + (0.647 - 0.762i)T \) |
| 11 | \( 1 + (-0.856 - 0.515i)T \) |
| 13 | \( 1 + (-0.561 - 0.827i)T \) |
| 17 | \( 1 + (0.647 + 0.762i)T \) |
| 19 | \( 1 + (0.907 - 0.419i)T \) |
| 23 | \( 1 + (-0.947 - 0.319i)T \) |
| 29 | \( 1 + (0.796 - 0.605i)T \) |
| 31 | \( 1 + (0.907 + 0.419i)T \) |
| 37 | \( 1 + (-0.370 - 0.928i)T \) |
| 41 | \( 1 + (-0.947 + 0.319i)T \) |
| 43 | \( 1 + (-0.856 + 0.515i)T \) |
| 47 | \( 1 + (-0.725 + 0.687i)T \) |
| 53 | \( 1 + (-0.161 + 0.986i)T \) |
| 61 | \( 1 + (0.796 + 0.605i)T \) |
| 67 | \( 1 + (-0.370 + 0.928i)T \) |
| 71 | \( 1 + (-0.725 + 0.687i)T \) |
| 73 | \( 1 + (0.976 - 0.214i)T \) |
| 79 | \( 1 + (0.468 - 0.883i)T \) |
| 83 | \( 1 + (-0.994 + 0.108i)T \) |
| 89 | \( 1 + (0.796 - 0.605i)T \) |
| 97 | \( 1 + (0.976 + 0.214i)T \) |
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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
−31.77723038037340695173967755432, −31.27451151672305971108794417864, −30.55830775295752221138460367230, −29.4529564538313870579993698723, −28.37954517095579003021794302538, −26.99268240480077789892650659783, −25.511002029074424857022877600648, −24.2636383574988280666478910512, −23.52340370884705107581523737064, −22.38365976067166399760877785403, −21.02270546502061721585136434954, −19.94639543798116168933591995641, −18.755924518556228230805185005, −18.21184281302567989808975945444, −15.67127119388775924443447203969, −14.63573800028213594785461996986, −13.78414665405196854400659824963, −12.06333155852948866380513100389, −11.76515818168751467870515103193, −9.9097939666370376639493381241, −8.0149477739548723960982444274, −6.78006029252296718247719221170, −5.07577448790977239672715969536, −3.20550969829395061568446471575, −2.02991432191879575303089581860,
3.159169833529844526087624592382, 4.42730207907219585457375824037, 5.36625637119392239479570135329, 7.76110188633408091627797822329, 8.30524476382904742344090577911, 10.35051772783521970209937069277, 11.7697195766729708593708005569, 13.2588952950647857625300660931, 14.38928448297996405978577904396, 15.53319992528961270753784073368, 16.32612030542425486594809013009, 17.46797054385587873022236143632, 19.771193081296756726901452887296, 20.641117511145924698239670564216, 21.53029041802279793907613479062, 22.91412413795104183165667447794, 23.92658183340109799616951958377, 24.89858807380180109122201203866, 26.41539046497927919862137029684, 26.96704000521899530528591937926, 28.30434420534345570276427384310, 30.10080333512958519712584125100, 31.07218309983260095949519120732, 32.12379787814029314034301846812, 32.61928218914903480511556029573