L(s) = 1 | + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (0.195 − 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (0.382 + 0.923i)23-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s − i·33-s + ⋯ |
L(s) = 1 | + (0.980 − 0.195i)3-s + (−0.555 − 0.831i)5-s + (0.923 + 0.382i)7-s + (0.923 − 0.382i)9-s + (0.195 − 0.980i)11-s + (−0.555 + 0.831i)13-s + (−0.707 − 0.707i)15-s + (0.707 − 0.707i)17-s + (−0.831 − 0.555i)19-s + (0.980 + 0.195i)21-s + (0.382 + 0.923i)23-s + (−0.382 + 0.923i)25-s + (0.831 − 0.555i)27-s + (0.195 + 0.980i)29-s − i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.427 - 0.903i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.305989759 - 1.460251910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.305989759 - 1.460251910i\) |
\(L(1)\) |
\(\approx\) |
\(1.508785290 - 0.4030528583i\) |
\(L(1)\) |
\(\approx\) |
\(1.508785290 - 0.4030528583i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 31 | \( 1 \) |
good | 3 | \( 1 + (0.980 - 0.195i)T \) |
| 5 | \( 1 + (-0.555 - 0.831i)T \) |
| 7 | \( 1 + (0.923 + 0.382i)T \) |
| 11 | \( 1 + (0.195 - 0.980i)T \) |
| 13 | \( 1 + (-0.555 + 0.831i)T \) |
| 17 | \( 1 + (0.707 - 0.707i)T \) |
| 19 | \( 1 + (-0.831 - 0.555i)T \) |
| 23 | \( 1 + (0.382 + 0.923i)T \) |
| 29 | \( 1 + (0.195 + 0.980i)T \) |
| 37 | \( 1 + (0.831 - 0.555i)T \) |
| 41 | \( 1 + (-0.382 - 0.923i)T \) |
| 43 | \( 1 + (0.980 + 0.195i)T \) |
| 47 | \( 1 + (-0.707 + 0.707i)T \) |
| 53 | \( 1 + (0.195 - 0.980i)T \) |
| 59 | \( 1 + (-0.555 - 0.831i)T \) |
| 61 | \( 1 + (0.980 - 0.195i)T \) |
| 67 | \( 1 + (0.980 - 0.195i)T \) |
| 71 | \( 1 + (-0.923 - 0.382i)T \) |
| 73 | \( 1 + (0.923 - 0.382i)T \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + (-0.831 - 0.555i)T \) |
| 89 | \( 1 + (-0.382 + 0.923i)T \) |
| 97 | \( 1 + iT \) |
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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
−18.635734854985288688676626781000, −18.130651441146288343203080358073, −17.23005898296402470352400557609, −16.68488225794302613831433593545, −15.47360707613356302866839297394, −15.091350785582976124438103793895, −14.58006730963416090261158736339, −14.23907940190740575611945944271, −13.14576831650496771325497088532, −12.526975481442260871549139087, −11.77233019887696566139068639637, −10.850491479791163842607253520569, −10.17218939570061996851931493582, −9.92647060620853515577726922729, −8.63361054469129430240102600366, −8.05185838437507007244890945874, −7.61076172444437760614749318570, −6.94529025400833181700309078473, −6.012550352018746054763623747265, −4.769663011993572544839529039971, −4.290795330383028001493622065768, −3.6010403261186682969700937814, −2.64692958033165443156979951914, −2.11278483066850641992753652566, −1.06413081443078088859291698663,
0.7518707929885622718816668041, 1.565017685961882852535420999595, 2.35866635353832940343540791446, 3.29502154368032030798622259680, 4.02646201072330692298754495819, 4.821718742439106922723401393497, 5.38741891783984672785619796569, 6.56930358237747099679734688038, 7.43834809815121793055520673672, 7.936361533632073033352993430499, 8.72835914785700137022992790096, 9.06096164653504968283952563626, 9.7610741901526268135040976271, 11.062316261098998588167214279083, 11.473045052643214955379663042473, 12.32653245219361402654528306285, 12.83718860149856007196748501509, 13.76975732868038251511335821842, 14.26770989497560967686019942206, 14.86026533869422722840062196232, 15.652564212537219186082917216177, 16.2343064830740335247334506174, 16.96184992960540953600845103731, 17.73603587170089229235942418295, 18.577315009765379658000049262633