L(s) = 1 | + (−0.974 − 0.225i)2-s + (0.576 − 0.816i)3-s + (0.898 + 0.439i)4-s + (−0.377 + 0.926i)5-s + (−0.746 + 0.665i)6-s + (0.715 − 0.699i)7-s + (−0.775 − 0.631i)8-s + (−0.334 − 0.942i)9-s + (0.576 − 0.816i)10-s + (−0.334 − 0.942i)11-s + (0.877 − 0.480i)12-s + (0.648 + 0.761i)13-s + (−0.854 + 0.519i)14-s + (0.538 + 0.842i)15-s + (0.613 + 0.789i)16-s + (0.460 + 0.887i)17-s + ⋯ |
L(s) = 1 | + (−0.974 − 0.225i)2-s + (0.576 − 0.816i)3-s + (0.898 + 0.439i)4-s + (−0.377 + 0.926i)5-s + (−0.746 + 0.665i)6-s + (0.715 − 0.699i)7-s + (−0.775 − 0.631i)8-s + (−0.334 − 0.942i)9-s + (0.576 − 0.816i)10-s + (−0.334 − 0.942i)11-s + (0.877 − 0.480i)12-s + (0.648 + 0.761i)13-s + (−0.854 + 0.519i)14-s + (0.538 + 0.842i)15-s + (0.613 + 0.789i)16-s + (0.460 + 0.887i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5319558444 - 1.166672623i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5319558444 - 1.166672623i\) |
\(L(1)\) |
\(\approx\) |
\(0.7754513945 - 0.3234874626i\) |
\(L(1)\) |
\(\approx\) |
\(0.7754513945 - 0.3234874626i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (-0.974 - 0.225i)T \) |
| 3 | \( 1 + (0.576 - 0.816i)T \) |
| 5 | \( 1 + (-0.377 + 0.926i)T \) |
| 7 | \( 1 + (0.715 - 0.699i)T \) |
| 11 | \( 1 + (-0.334 - 0.942i)T \) |
| 13 | \( 1 + (0.648 + 0.761i)T \) |
| 17 | \( 1 + (0.460 + 0.887i)T \) |
| 19 | \( 1 + (-0.995 - 0.0909i)T \) |
| 23 | \( 1 + (0.0682 + 0.997i)T \) |
| 29 | \( 1 + (0.334 - 0.942i)T \) |
| 31 | \( 1 + (0.934 - 0.356i)T \) |
| 37 | \( 1 + (0.648 + 0.761i)T \) |
| 41 | \( 1 + (-0.203 - 0.979i)T \) |
| 43 | \( 1 + (0.998 - 0.0455i)T \) |
| 47 | \( 1 + (-0.538 - 0.842i)T \) |
| 53 | \( 1 + (-0.829 + 0.557i)T \) |
| 59 | \( 1 + (-0.829 - 0.557i)T \) |
| 61 | \( 1 + (-0.949 - 0.313i)T \) |
| 67 | \( 1 + (0.775 - 0.631i)T \) |
| 71 | \( 1 + (0.682 + 0.730i)T \) |
| 73 | \( 1 + (-0.829 - 0.557i)T \) |
| 79 | \( 1 + (0.829 + 0.557i)T \) |
| 83 | \( 1 + (0.746 - 0.665i)T \) |
| 89 | \( 1 + (0.158 + 0.987i)T \) |
| 97 | \( 1 + T \) |
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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
−21.23872454687245187755521189268, −20.94188240036393011136503053813, −20.27401135746459331859547921075, −19.6445403773664770657397845250, −18.64549111606667510619887594456, −17.8946163185846156227239598983, −17.07443958503875272665048214427, −16.13233072363893216165521218044, −15.732217154059595870093487942432, −14.938456158936077535583844059638, −14.3358049246652879005108379308, −12.886312040724334698071213854060, −12.11949928828807027243023059686, −11.102023246293208847464535905103, −10.40673150697462033985933708881, −9.452669781102968589382774009604, −8.80867239490320186996069125106, −8.13567966493078283219553728415, −7.62129521945474403164117500355, −6.121312897493227853700399508736, −5.05684644889701822914728777842, −4.554972362087866741102995517331, −3.03854331907718752949078568265, −2.157216786008764501989138285994, −1.03063687707575837074497174655,
0.38482911181938222689830452747, 1.426603055909128783816643352413, 2.29457363871503219426888798856, 3.33759346902201944632576302787, 4.02209237466385617352407546514, 6.12822365629244409386069098254, 6.56816777532613253331894449021, 7.71386575578177036592072479476, 8.0041350005422758558128947225, 8.812762599974460920525670686915, 9.97338117982419228341965017245, 10.904257157660747817789188086750, 11.3607916048177674434000027594, 12.21653287553013906987964677266, 13.44102198457725751623806963388, 14.01465091801952407278301849301, 15.00475374690659124377135712740, 15.64839070796884784554128959980, 16.91750712266540509169890118735, 17.4476790074449639661094467840, 18.35192338808564801710579911000, 19.130484810144238189181468587166, 19.20313894222628941647737616740, 20.277123891369866952013973158886, 21.183378789663784177196850593815