L(s) = 1 | + (−0.958 − 0.285i)2-s + (−0.168 − 0.985i)3-s + (0.836 + 0.548i)4-s + (−0.443 + 0.896i)5-s + (−0.120 + 0.992i)6-s + (0.485 − 0.873i)7-s + (−0.644 − 0.764i)8-s + (−0.943 + 0.331i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.981 + 0.192i)13-s + (−0.715 + 0.698i)14-s + (0.958 + 0.285i)15-s + (0.399 + 0.916i)16-s + (0.748 − 0.663i)17-s + (0.998 − 0.0483i)18-s + ⋯ |
L(s) = 1 | + (−0.958 − 0.285i)2-s + (−0.168 − 0.985i)3-s + (0.836 + 0.548i)4-s + (−0.443 + 0.896i)5-s + (−0.120 + 0.992i)6-s + (0.485 − 0.873i)7-s + (−0.644 − 0.764i)8-s + (−0.943 + 0.331i)9-s + (0.681 − 0.732i)10-s + (0.399 − 0.916i)12-s + (0.981 + 0.192i)13-s + (−0.715 + 0.698i)14-s + (0.958 + 0.285i)15-s + (0.399 + 0.916i)16-s + (0.748 − 0.663i)17-s + (0.998 − 0.0483i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.215 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5903240968 - 0.7344691750i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5903240968 - 0.7344691750i\) |
\(L(1)\) |
\(\approx\) |
\(0.6029431878 - 0.2273300131i\) |
\(L(1)\) |
\(\approx\) |
\(0.6029431878 - 0.2273300131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 131 | \( 1 \) |
good | 2 | \( 1 + (-0.958 - 0.285i)T \) |
| 3 | \( 1 + (-0.168 - 0.985i)T \) |
| 5 | \( 1 + (-0.443 + 0.896i)T \) |
| 7 | \( 1 + (0.485 - 0.873i)T \) |
| 13 | \( 1 + (0.981 + 0.192i)T \) |
| 17 | \( 1 + (0.748 - 0.663i)T \) |
| 19 | \( 1 + (-0.399 + 0.916i)T \) |
| 23 | \( 1 + (-0.644 + 0.764i)T \) |
| 29 | \( 1 + (-0.0241 + 0.999i)T \) |
| 31 | \( 1 + (-0.644 - 0.764i)T \) |
| 37 | \( 1 + (0.262 - 0.964i)T \) |
| 41 | \( 1 + (-0.748 - 0.663i)T \) |
| 43 | \( 1 + (0.715 + 0.698i)T \) |
| 47 | \( 1 + (-0.958 - 0.285i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.215 - 0.976i)T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + (-0.715 + 0.698i)T \) |
| 71 | \( 1 + (-0.644 - 0.764i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (0.443 - 0.896i)T \) |
| 83 | \( 1 + (0.262 + 0.964i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (-0.485 + 0.873i)T \) |
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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
−20.84392750996925640532169665447, −19.989377617579674214828870728908, −19.29819338263308165886469694556, −18.37200902492483584532939341664, −17.63222847935716541444203315355, −16.8859640407016952188753155343, −16.25139589538907494781756912082, −15.59904913348277023658752248434, −15.10700879132628499507658814739, −14.323084243177392047744367445279, −12.98189997101686769392368096422, −11.9210689031930700301568376179, −11.48091305366841628600506512910, −10.62654315594467270149383670460, −9.83092457488395770562802389166, −8.94021818488731523961832422006, −8.46081986579098994997825532240, −7.97726821875605051229598826523, −6.48343068420715168045804698333, −5.704671932931779349561061131627, −5.04420244506193887192182595996, −4.07002338722703498767774864417, −2.956266775409348855462285446, −1.775629226402319702779647357059, −0.68456487607794465823967983410,
0.38298612731927849947875386345, 1.35794102319239459783495064494, 2.102111732697491940559719509774, 3.297543408019387521791008337026, 3.89281330144227564929702241118, 5.63044130759658401348247798553, 6.48103333043011284192781808719, 7.251013670947987167999344955706, 7.759165385643385950193032282038, 8.35582422055496036776893099017, 9.51823775605328891188345182704, 10.559146468548544016636578694614, 11.02981878526277665028449049248, 11.691198174015328524897704739023, 12.413485131622966912219883562374, 13.461237219350587962628839548087, 14.21451949864635172878705440805, 14.94218263434884769570006507930, 16.253579953992994877293119367527, 16.5577788627786094080848291724, 17.81668742835758343998340980232, 17.96570192203936002517991076814, 18.85680420423311180480962071990, 19.268344919504703158855540069346, 20.21071154047500768866319490717