| L(s) = 1 | + (0.251 + 2.38i)2-s + (0.440 + 2.31i)3-s + (−3.68 + 0.783i)4-s + (0.526 − 3.56i)5-s + (−5.40 + 1.63i)6-s + (−2.36 + 2.41i)7-s + (−1.31 − 4.03i)8-s + (−2.35 + 0.932i)9-s + (8.65 + 0.362i)10-s + (1.81 + 1.56i)11-s + (−3.43 − 8.16i)12-s + (3.34 + 2.02i)13-s + (−6.35 − 5.03i)14-s + (8.47 − 0.355i)15-s + (2.42 − 1.08i)16-s + (0.693 − 2.48i)17-s + ⋯ |
| L(s) = 1 | + (0.177 + 1.68i)2-s + (0.254 + 1.33i)3-s + (−1.84 + 0.391i)4-s + (0.235 − 1.59i)5-s + (−2.20 + 0.666i)6-s + (−0.893 + 0.912i)7-s + (−0.463 − 1.42i)8-s + (−0.784 + 0.310i)9-s + (2.73 + 0.114i)10-s + (0.546 + 0.471i)11-s + (−0.991 − 2.35i)12-s + (0.926 + 0.561i)13-s + (−1.69 − 1.34i)14-s + (2.18 − 0.0917i)15-s + (0.606 − 0.270i)16-s + (0.168 − 0.602i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0391689 + 1.19162i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0391689 + 1.19162i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (9.04 - 8.32i)T \) |
| good | 2 | \( 1 + (-0.251 - 2.38i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-0.440 - 2.31i)T + (-2.78 + 1.10i)T^{2} \) |
| 5 | \( 1 + (-0.526 + 3.56i)T + (-4.78 - 1.44i)T^{2} \) |
| 7 | \( 1 + (2.36 - 2.41i)T + (-0.146 - 6.99i)T^{2} \) |
| 11 | \( 1 + (-1.81 - 1.56i)T + (1.60 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.34 - 2.02i)T + (6.02 + 11.5i)T^{2} \) |
| 17 | \( 1 + (-0.693 + 2.48i)T + (-14.5 - 8.80i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 4.96i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-4.83 - 5.37i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-0.966 - 2.05i)T + (-18.4 + 22.3i)T^{2} \) |
| 31 | \( 1 + (2.89 + 7.79i)T + (-23.4 + 20.2i)T^{2} \) |
| 37 | \( 1 + (-0.0376 + 1.79i)T + (-36.9 - 1.54i)T^{2} \) |
| 41 | \( 1 + (8.74 + 8.21i)T + (2.57 + 40.9i)T^{2} \) |
| 43 | \( 1 + (2.64 + 2.70i)T + (-0.900 + 42.9i)T^{2} \) |
| 47 | \( 1 + (0.135 - 0.578i)T + (-42.0 - 20.8i)T^{2} \) |
| 53 | \( 1 + (-6.50 + 0.821i)T + (51.3 - 13.1i)T^{2} \) |
| 59 | \( 1 + (-0.610 - 0.443i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.22 + 2.86i)T + (47.8 - 37.8i)T^{2} \) |
| 67 | \( 1 + (5.70 + 2.26i)T + (48.8 + 45.8i)T^{2} \) |
| 71 | \( 1 + (-3.74 - 13.4i)T + (-60.7 + 36.7i)T^{2} \) |
| 73 | \( 1 + (8.50 + 2.18i)T + (63.9 + 35.1i)T^{2} \) |
| 79 | \( 1 + (3.49 + 4.21i)T + (-14.8 + 77.6i)T^{2} \) |
| 83 | \( 1 + (-0.151 + 2.40i)T + (-82.3 - 10.4i)T^{2} \) |
| 89 | \( 1 + (0.0793 - 0.0393i)T + (53.8 - 70.8i)T^{2} \) |
| 97 | \( 1 + (-2.26 + 1.79i)T + (22.1 - 94.4i)T^{2} \) |
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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
−13.62462897815562781341098255537, −12.99038943560076591592422346508, −11.65216085674030450136140946188, −9.620393420782862009823848318838, −9.068764232941007452807213742737, −8.752308123303415974547686282597, −7.00516215161949135122242131394, −5.61239823894532882896639654007, −4.98175583833968316501524727745, −3.86536479205801138264366534727,
1.33218640350376902902463757004, 2.96097909159327351878519471936, 3.60526878242936867321884530123, 6.28098171297237632408221283988, 7.00072322522666565357927035259, 8.449716198298960566520256127120, 10.10546335907981928448872226589, 10.52447578167607715534839677792, 11.51438404101549779744485355367, 12.58505605932512291199591255952
