| L(s) = 1 | + (0.287 − 2.73i)2-s + (0.548 − 0.515i)3-s + (−5.46 − 1.16i)4-s + (−0.949 + 0.471i)5-s + (−1.25 − 1.65i)6-s + (1.63 − 3.13i)7-s + (−3.05 + 9.40i)8-s + (−0.152 + 2.42i)9-s + (1.01 + 2.73i)10-s + (0.863 − 3.67i)11-s + (−3.59 + 2.17i)12-s + (−1.19 − 0.100i)13-s + (−8.10 − 5.38i)14-s + (−0.278 + 0.748i)15-s + (14.6 + 6.53i)16-s + (6.40 + 0.268i)17-s + ⋯ |
| L(s) = 1 | + (0.203 − 1.93i)2-s + (0.316 − 0.297i)3-s + (−2.73 − 0.580i)4-s + (−0.424 + 0.210i)5-s + (−0.511 − 0.674i)6-s + (0.618 − 1.18i)7-s + (−1.08 + 3.32i)8-s + (−0.0509 + 0.809i)9-s + (0.321 + 0.865i)10-s + (0.260 − 1.10i)11-s + (−1.03 + 0.629i)12-s + (−0.331 − 0.0278i)13-s + (−2.16 − 1.43i)14-s + (−0.0718 + 0.193i)15-s + (3.66 + 1.63i)16-s + (1.55 + 0.0650i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0421i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0232395 + 1.10335i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0232395 + 1.10335i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (10.5 + 6.34i)T \) |
| good | 2 | \( 1 + (-0.287 + 2.73i)T + (-1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.548 + 0.515i)T + (0.188 - 2.99i)T^{2} \) |
| 5 | \( 1 + (0.949 - 0.471i)T + (3.02 - 3.98i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 3.13i)T + (-3.99 - 5.74i)T^{2} \) |
| 11 | \( 1 + (-0.863 + 3.67i)T + (-9.85 - 4.89i)T^{2} \) |
| 13 | \( 1 + (1.19 + 0.100i)T + (12.8 + 2.16i)T^{2} \) |
| 17 | \( 1 + (-6.40 - 0.268i)T + (16.9 + 1.42i)T^{2} \) |
| 19 | \( 1 + (0.616 + 1.89i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-2.17 + 2.41i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.230 + 1.20i)T + (-26.9 + 10.6i)T^{2} \) |
| 31 | \( 1 + (1.09 + 0.871i)T + (7.07 + 30.1i)T^{2} \) |
| 37 | \( 1 + (5.99 - 8.62i)T + (-12.8 - 34.6i)T^{2} \) |
| 41 | \( 1 + (-10.3 + 1.31i)T + (39.7 - 10.1i)T^{2} \) |
| 43 | \( 1 + (0.771 + 1.47i)T + (-24.5 + 35.3i)T^{2} \) |
| 47 | \( 1 + (2.82 - 6.72i)T + (-32.8 - 33.5i)T^{2} \) |
| 53 | \( 1 + (-3.85 + 2.11i)T + (28.3 - 44.7i)T^{2} \) |
| 59 | \( 1 + (1.30 - 0.951i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.98 - 1.20i)T + (50.8 - 33.7i)T^{2} \) |
| 67 | \( 1 + (-0.788 - 12.5i)T + (-66.4 + 8.39i)T^{2} \) |
| 71 | \( 1 + (-9.81 + 0.411i)T + (70.7 - 5.94i)T^{2} \) |
| 73 | \( 1 + (0.862 + 1.35i)T + (-31.0 + 66.0i)T^{2} \) |
| 79 | \( 1 + (-4.81 - 1.90i)T + (57.5 + 54.0i)T^{2} \) |
| 83 | \( 1 + (2.47 + 0.634i)T + (72.7 + 39.9i)T^{2} \) |
| 89 | \( 1 + (11.5 - 11.7i)T + (-1.86 - 88.9i)T^{2} \) |
| 97 | \( 1 + (-4.78 + 3.18i)T + (37.5 - 89.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.36765972448412502502447901112, −11.26926103090624624336182808883, −10.83738718655720712064955389071, −9.867484690902040438023088411646, −8.502189854693673353869931884292, −7.64552471399217809741145753924, −5.26899883943876388072794817045, −4.04413790848670809109380339366, −2.94413720894496131368920569155, −1.19230172882386311492006946707,
3.77852209664194131867749836154, 4.99076815085710619144702800542, 5.91894055324662664253330504648, 7.26176810915714654466035021758, 8.123997653126762874909016410036, 9.073140506075846488571249406048, 9.760443817002114112407496227606, 12.19718893296605514492292315164, 12.45659541170473933356706053708, 14.18789220634350794246894930018
