| L(s) = 1 | + (2.57 − 0.546i)2-s + (−0.539 + 0.213i)3-s + (4.48 − 1.99i)4-s + (−2.66 − 0.805i)5-s + (−1.27 + 0.845i)6-s + (0.00307 + 0.146i)7-s + (6.20 − 4.50i)8-s + (−1.94 + 1.82i)9-s + (−7.29 − 0.612i)10-s + (0.367 + 2.49i)11-s + (−1.99 + 2.03i)12-s + (1.83 + 3.50i)13-s + (0.0882 + 0.376i)14-s + (1.61 − 0.135i)15-s + (6.90 − 7.67i)16-s + (−1.47 − 0.891i)17-s + ⋯ |
| L(s) = 1 | + (1.81 − 0.386i)2-s + (−0.311 + 0.123i)3-s + (2.24 − 0.999i)4-s + (−1.19 − 0.360i)5-s + (−0.519 + 0.344i)6-s + (0.00116 + 0.0555i)7-s + (2.19 − 1.59i)8-s + (−0.647 + 0.607i)9-s + (−2.30 − 0.193i)10-s + (0.110 + 0.750i)11-s + (−0.576 + 0.588i)12-s + (0.508 + 0.972i)13-s + (0.0235 + 0.100i)14-s + (0.416 − 0.0349i)15-s + (1.72 − 1.91i)16-s + (−0.356 − 0.216i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.17684 - 0.582913i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.17684 - 0.582913i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (10.4 - 6.40i)T \) |
| good | 2 | \( 1 + (-2.57 + 0.546i)T + (1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (0.539 - 0.213i)T + (2.18 - 2.05i)T^{2} \) |
| 5 | \( 1 + (2.66 + 0.805i)T + (4.16 + 2.76i)T^{2} \) |
| 7 | \( 1 + (-0.00307 - 0.146i)T + (-6.99 + 0.293i)T^{2} \) |
| 11 | \( 1 + (-0.367 - 2.49i)T + (-10.5 + 3.17i)T^{2} \) |
| 13 | \( 1 + (-1.83 - 3.50i)T + (-7.41 + 10.6i)T^{2} \) |
| 17 | \( 1 + (1.47 + 0.891i)T + (7.87 + 15.0i)T^{2} \) |
| 19 | \( 1 + (3.49 + 2.53i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.686 + 6.53i)T + (-22.4 - 4.78i)T^{2} \) |
| 29 | \( 1 + (-5.91 + 7.14i)T + (-5.43 - 28.4i)T^{2} \) |
| 31 | \( 1 + (6.10 - 5.27i)T + (4.52 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-2.86 - 0.120i)T + (36.8 + 3.09i)T^{2} \) |
| 41 | \( 1 + (0.558 + 8.88i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (0.162 - 7.75i)T + (-42.9 - 1.80i)T^{2} \) |
| 47 | \( 1 + (-9.98 - 4.95i)T + (28.4 + 37.4i)T^{2} \) |
| 53 | \( 1 + (11.2 - 2.88i)T + (46.4 - 25.5i)T^{2} \) |
| 59 | \( 1 + (-1.75 - 5.38i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-0.408 + 0.323i)T + (13.9 - 59.3i)T^{2} \) |
| 67 | \( 1 + (-6.59 - 6.18i)T + (4.20 + 66.8i)T^{2} \) |
| 71 | \( 1 + (2.98 - 1.80i)T + (32.8 - 62.9i)T^{2} \) |
| 73 | \( 1 + (7.58 + 4.17i)T + (39.1 + 61.6i)T^{2} \) |
| 79 | \( 1 + (0.213 - 1.11i)T + (-73.4 - 29.0i)T^{2} \) |
| 83 | \( 1 + (11.8 + 1.49i)T + (80.3 + 20.6i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 2.07i)T + (-23.9 - 85.7i)T^{2} \) |
| 97 | \( 1 + (-2.75 + 11.7i)T + (-86.8 - 43.1i)T^{2} \) |
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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
−12.75763468465651503169101951637, −12.05121277650090431583181673178, −11.32352519695163309209506418970, −10.60432574954409071436229595132, −8.665871633250289994830777006890, −7.19068395807053212296463327090, −6.10457133405463393171887809560, −4.62650520048534360156272465968, −4.23670116987199852696950726811, −2.49205083370583140834778124776,
3.21246362399770002363493924995, 3.88121509527952542694753822568, 5.45607933212982872396579125381, 6.30390341267463563689157485368, 7.41924707825812721788089494347, 8.461160835650740313921253691851, 10.86754078338860632190487391274, 11.38479347101126048378636307874, 12.25494711985173963836310471805, 13.05798413204608304406458098722
