| L(s) = 1 | + (−0.497 − 0.552i)2-s + (1.93 + 2.33i)3-s + (0.151 − 1.43i)4-s + (0.151 + 0.130i)5-s + (0.329 − 2.23i)6-s + (1.65 − 3.93i)7-s + (−2.07 + 1.50i)8-s + (−1.16 + 6.08i)9-s + (−0.00311 − 0.148i)10-s + (0.122 + 0.328i)11-s + (3.65 − 2.42i)12-s + (1.16 + 4.17i)13-s + (−2.99 + 1.04i)14-s + (−0.0126 + 0.605i)15-s + (−0.962 − 0.204i)16-s + (−2.47 + 3.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.352 − 0.391i)2-s + (1.11 + 1.34i)3-s + (0.0755 − 0.719i)4-s + (0.0676 + 0.0583i)5-s + (0.134 − 0.911i)6-s + (0.624 − 1.48i)7-s + (−0.733 + 0.532i)8-s + (−0.386 + 2.02i)9-s + (−0.000984 − 0.0469i)10-s + (0.0368 + 0.0991i)11-s + (1.05 − 0.700i)12-s + (0.323 + 1.15i)13-s + (−0.800 + 0.278i)14-s + (−0.00327 + 0.156i)15-s + (−0.240 − 0.0511i)16-s + (−0.601 + 0.792i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00760i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.32097 + 0.00502214i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.32097 + 0.00502214i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-6.04 - 10.7i)T \) |
| good | 2 | \( 1 + (0.497 + 0.552i)T + (-0.209 + 1.98i)T^{2} \) |
| 3 | \( 1 + (-1.93 - 2.33i)T + (-0.562 + 2.94i)T^{2} \) |
| 5 | \( 1 + (-0.151 - 0.130i)T + (0.730 + 4.94i)T^{2} \) |
| 7 | \( 1 + (-1.65 + 3.93i)T + (-4.89 - 5.00i)T^{2} \) |
| 11 | \( 1 + (-0.122 - 0.328i)T + (-8.32 + 7.18i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 4.17i)T + (-11.1 + 6.73i)T^{2} \) |
| 17 | \( 1 + (2.47 - 3.26i)T + (-4.57 - 16.3i)T^{2} \) |
| 19 | \( 1 + (4.21 + 3.06i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.933 + 0.415i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (4.04 + 6.37i)T + (-12.3 + 26.2i)T^{2} \) |
| 31 | \( 1 + (-2.24 - 3.23i)T + (-10.8 + 29.0i)T^{2} \) |
| 37 | \( 1 + (3.94 - 4.02i)T + (-0.774 - 36.9i)T^{2} \) |
| 41 | \( 1 + (3.90 - 1.54i)T + (29.8 - 28.0i)T^{2} \) |
| 43 | \( 1 + (1.84 + 4.38i)T + (-30.0 + 30.7i)T^{2} \) |
| 47 | \( 1 + (-6.58 - 5.21i)T + (10.7 + 45.7i)T^{2} \) |
| 53 | \( 1 + (-0.699 + 11.1i)T + (-52.5 - 6.64i)T^{2} \) |
| 59 | \( 1 + (-3.91 - 12.0i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (3.70 + 0.626i)T + (57.6 + 20.0i)T^{2} \) |
| 67 | \( 1 + (0.975 + 5.11i)T + (-62.2 + 24.6i)T^{2} \) |
| 71 | \( 1 + (-3.88 - 5.11i)T + (-19.0 + 68.3i)T^{2} \) |
| 73 | \( 1 + (-15.9 + 2.01i)T + (70.7 - 18.1i)T^{2} \) |
| 79 | \( 1 + (0.231 + 0.492i)T + (-50.3 + 60.8i)T^{2} \) |
| 83 | \( 1 + (9.40 + 8.82i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (1.30 - 5.54i)T + (-79.7 - 39.5i)T^{2} \) |
| 97 | \( 1 + (-1.62 - 0.566i)T + (76.0 + 60.2i)T^{2} \) |
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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
−13.55272085065914343865111323062, −11.41657362838687342010650808221, −10.61171337452691243969028322361, −10.11954862463508098430913119045, −9.049919887093076882602049328362, −8.274495390275724960602463566892, −6.65238502842820651949492559651, −4.69531130736325757652537013408, −3.97596371145101177741478606263, −2.09694423124282416773004718832,
2.13670585083281419144339082009, 3.26960072158419494887109972151, 5.70696073659236784945678981645, 6.97011456154026610317999962647, 7.947108185451963638983874538300, 8.593290076438986206203150914205, 9.203318050264638672101140615836, 11.33280613973154938385387202478, 12.45090854899600482618821161503, 12.79641901658492111735247956424
