L(s) = 1 | + (−0.995 + 0.0950i)2-s + (−0.5 + 0.866i)3-s + (0.981 − 0.189i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 − 0.866i)9-s + (−0.327 + 0.945i)12-s + (0.654 + 0.755i)13-s + (0.928 − 0.371i)16-s + (0.888 − 0.458i)17-s + (0.580 + 0.814i)18-s + (−0.888 − 0.458i)19-s + (−0.928 + 0.371i)23-s + (0.235 − 0.971i)24-s + (−0.723 − 0.690i)26-s + 27-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0950i)2-s + (−0.5 + 0.866i)3-s + (0.981 − 0.189i)4-s + (0.415 − 0.909i)6-s + (−0.959 + 0.281i)8-s + (−0.5 − 0.866i)9-s + (−0.327 + 0.945i)12-s + (0.654 + 0.755i)13-s + (0.928 − 0.371i)16-s + (0.888 − 0.458i)17-s + (0.580 + 0.814i)18-s + (−0.888 − 0.458i)19-s + (−0.928 + 0.371i)23-s + (0.235 − 0.971i)24-s + (−0.723 − 0.690i)26-s + 27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3524573272 - 0.1957271250i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3524573272 - 0.1957271250i\) |
\(L(1)\) |
\(\approx\) |
\(0.5180770460 + 0.1292411409i\) |
\(L(1)\) |
\(\approx\) |
\(0.5180770460 + 0.1292411409i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.995 + 0.0950i)T \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.654 + 0.755i)T \) |
| 17 | \( 1 + (0.888 - 0.458i)T \) |
| 19 | \( 1 + (-0.888 - 0.458i)T \) |
| 23 | \( 1 + (-0.928 + 0.371i)T \) |
| 29 | \( 1 + (-0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.327 + 0.945i)T \) |
| 37 | \( 1 + (-0.981 - 0.189i)T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (-0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.580 - 0.814i)T \) |
| 53 | \( 1 + (-0.928 - 0.371i)T \) |
| 59 | \( 1 + (0.995 + 0.0950i)T \) |
| 61 | \( 1 + (0.580 - 0.814i)T \) |
| 67 | \( 1 + (-0.580 - 0.814i)T \) |
| 71 | \( 1 + (0.841 - 0.540i)T \) |
| 73 | \( 1 + (0.786 + 0.618i)T \) |
| 79 | \( 1 + (-0.235 - 0.971i)T \) |
| 83 | \( 1 + (0.142 + 0.989i)T \) |
| 89 | \( 1 + (-0.0475 + 0.998i)T \) |
| 97 | \( 1 + (-0.959 + 0.281i)T \) |
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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
−18.58938569817384441297195497185, −17.79242874744779774560748127190, −17.27359773500232146404285584117, −16.67493938882603548133921382666, −16.080369067089660793318238576021, −15.20975849936183209055981891397, −14.53217106296388622660822171282, −13.55293489143770229742796324534, −12.79067160584868020368934499964, −12.30278875593660735624373617983, −11.56365991321041531822900317507, −10.918834680933729555298803000803, −10.280418844538014971650332658136, −9.66331589999680114686417262808, −8.40838874534076302049105479481, −8.21533507828744423464069253336, −7.51189380662982811931063779974, −6.670101710376854871158001195745, −5.93624289439414144448265809457, −5.62430850356542893208590710185, −4.19566123453369023233101428108, −3.25630125500657325858546880412, −2.33777910464252583555270298648, −1.63216522282679940974755467969, −0.84925026323261694492158242588,
0.210162597934500472375052319051, 1.35739927766269942858794224625, 2.22267986405139767636858431706, 3.359890021367976635798748636380, 3.881394739808435711613217242089, 5.0590312127236162740277553446, 5.62037365764133412759621840462, 6.53300182498752714953408796573, 6.9685200743547483669726997667, 8.085911407677346468186383757464, 8.70485197138693151697917557944, 9.38270210261163187898060455406, 9.922049043716239109588419146439, 10.74959391453212215375131375002, 11.09696373535987091081959948254, 11.98150290109861379992161364849, 12.37763776315880882633196783762, 13.70341664336836719995626883333, 14.43919468739212084680303489181, 15.11517331166324015029943214601, 15.88002212979333105577691794603, 16.24614254273129257039094059104, 16.89954708611119891654353574620, 17.51614816661384637611207558052, 18.15617639375939599247602123216