L(s) = 1 | + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯ |
L(s) = 1 | + (−0.442 − 0.896i)3-s + (−0.946 + 0.321i)5-s + (−0.608 + 0.793i)9-s + (0.0654 + 0.997i)11-s + (−0.195 − 0.980i)13-s + (0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.659 − 0.751i)19-s + (−0.793 − 0.608i)23-s + (0.793 − 0.608i)25-s + (0.980 + 0.195i)27-s + (−0.831 − 0.555i)29-s + (0.866 + 0.5i)31-s + (0.866 − 0.5i)33-s + (−0.321 − 0.946i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.453 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2203231858 + 0.1351471190i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2203231858 + 0.1351471190i\) |
\(L(1)\) |
\(\approx\) |
\(0.5975485460 - 0.1849666353i\) |
\(L(1)\) |
\(\approx\) |
\(0.5975485460 - 0.1849666353i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.442 - 0.896i)T \) |
| 5 | \( 1 + (-0.946 + 0.321i)T \) |
| 11 | \( 1 + (0.0654 + 0.997i)T \) |
| 13 | \( 1 + (-0.195 - 0.980i)T \) |
| 17 | \( 1 + (-0.258 - 0.965i)T \) |
| 19 | \( 1 + (-0.659 - 0.751i)T \) |
| 23 | \( 1 + (-0.793 - 0.608i)T \) |
| 29 | \( 1 + (-0.831 - 0.555i)T \) |
| 31 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + (-0.321 - 0.946i)T \) |
| 41 | \( 1 + (0.923 - 0.382i)T \) |
| 43 | \( 1 + (0.555 + 0.831i)T \) |
| 47 | \( 1 + (-0.965 - 0.258i)T \) |
| 53 | \( 1 + (-0.0654 - 0.997i)T \) |
| 59 | \( 1 + (-0.751 - 0.659i)T \) |
| 61 | \( 1 + (-0.997 - 0.0654i)T \) |
| 67 | \( 1 + (-0.442 - 0.896i)T \) |
| 71 | \( 1 + (-0.382 + 0.923i)T \) |
| 73 | \( 1 + (0.991 + 0.130i)T \) |
| 79 | \( 1 + (-0.258 + 0.965i)T \) |
| 83 | \( 1 + (-0.980 + 0.195i)T \) |
| 89 | \( 1 + (0.130 + 0.991i)T \) |
| 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
−21.565992891055268381546663486026, −21.01038286152399810015402981685, −20.05241590047928813164166052076, −19.28953162757487092340855375486, −18.626407266522788859956885482957, −17.32334456504195555375516240950, −16.68606420717123285079624569557, −16.152602467936988933833034325966, −15.315638221112706166005351710733, −14.64228365148446720812263263980, −13.66914534923859133671259634612, −12.46406429350389920466215465122, −11.771316363992939952745236656464, −11.097473193975963451866396051279, −10.34981363432073494723952175853, −9.24288840587136407770012982750, −8.57513986841380861226636305919, −7.72072920450199178820487812359, −6.3958257615630157777635093971, −5.738294965441943831515900027629, −4.49226570962520486457560789001, −4.01052539138398922190849960503, −3.13464883979251155246980539841, −1.496967038470941465664896182022, −0.09622258367748272216968697284,
0.63814383948518159773855599696, 2.12235044129866398688271162210, 2.91524271463680258434330780081, 4.28550211149641046695511430945, 5.07249234091161426822957776044, 6.26417690040631398964253593449, 7.07522770430050637195938071255, 7.6864765395123267533592197606, 8.44736221380307609829030148975, 9.706942162301514352037310246392, 10.790796885355136325143980420607, 11.37945721966654204597227835235, 12.37287555949877097496548619935, 12.698081517487534027763230530726, 13.81877556017570189354055238382, 14.72213385019815741273525651408, 15.52638375196100156127969893356, 16.28382254467222937291101358545, 17.41118750178831659648987408959, 17.922894532282737811552968752572, 18.65096359001594540429540294893, 19.68399935444660908313472345320, 19.90822959736331551991884312186, 20.99739535339810536277140420321, 22.38746813063146994415294704547