| L(s) = 1 | + (0.0971 − 0.0432i)2-s + (−2.79 − 1.53i)3-s + (−1.33 + 1.47i)4-s + (2.31 − 0.0970i)5-s + (−0.338 − 0.0284i)6-s + (−2.95 + 2.34i)7-s + (−0.131 + 0.403i)8-s + (3.84 + 6.06i)9-s + (0.220 − 0.109i)10-s + (−0.101 + 4.82i)11-s + (5.99 − 2.08i)12-s + (−4.30 − 1.30i)13-s + (−0.185 + 0.355i)14-s + (−6.62 − 3.28i)15-s + (−0.411 − 3.91i)16-s + (0.156 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (0.0687 − 0.0305i)2-s + (−1.61 − 0.887i)3-s + (−0.665 + 0.738i)4-s + (1.03 − 0.0434i)5-s + (−0.138 − 0.0115i)6-s + (−1.11 + 0.884i)7-s + (−0.0463 + 0.142i)8-s + (1.28 + 2.02i)9-s + (0.0698 − 0.0346i)10-s + (−0.0304 + 1.45i)11-s + (1.72 − 0.602i)12-s + (−1.19 − 0.360i)13-s + (−0.0496 + 0.0949i)14-s + (−1.71 − 0.849i)15-s + (−0.102 − 0.977i)16-s + (0.0378 − 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227300 + 0.309968i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227300 + 0.309968i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-10.7 - 6.03i)T \) |
| good | 2 | \( 1 + (-0.0971 + 0.0432i)T + (1.33 - 1.48i)T^{2} \) |
| 3 | \( 1 + (2.79 + 1.53i)T + (1.60 + 2.53i)T^{2} \) |
| 5 | \( 1 + (-2.31 + 0.0970i)T + (4.98 - 0.418i)T^{2} \) |
| 7 | \( 1 + (2.95 - 2.34i)T + (1.59 - 6.81i)T^{2} \) |
| 11 | \( 1 + (0.101 - 4.82i)T + (-10.9 - 0.460i)T^{2} \) |
| 13 | \( 1 + (4.30 + 1.30i)T + (10.8 + 7.19i)T^{2} \) |
| 17 | \( 1 + (-0.156 + 1.05i)T + (-16.2 - 4.91i)T^{2} \) |
| 19 | \( 1 + (-0.938 - 2.88i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (2.74 + 0.584i)T + (21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (1.75 + 0.221i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (-2.80 + 2.86i)T + (-0.649 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-2.64 - 11.2i)T + (-33.1 + 16.4i)T^{2} \) |
| 41 | \( 1 + (1.69 + 3.60i)T + (-26.1 + 31.5i)T^{2} \) |
| 43 | \( 1 + (0.299 + 0.237i)T + (9.81 + 41.8i)T^{2} \) |
| 47 | \( 1 + (1.73 - 2.49i)T + (-16.3 - 44.0i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 6.44i)T + (-49.2 + 19.5i)T^{2} \) |
| 59 | \( 1 + (-1.48 + 1.07i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.37 - 3.25i)T + (28.2 + 54.0i)T^{2} \) |
| 67 | \( 1 + (-3.00 + 4.72i)T + (-28.5 - 60.6i)T^{2} \) |
| 71 | \( 1 + (1.03 + 7.02i)T + (-67.9 + 20.5i)T^{2} \) |
| 73 | \( 1 + (-4.38 - 1.73i)T + (53.2 + 49.9i)T^{2} \) |
| 79 | \( 1 + (-2.03 + 0.522i)T + (69.2 - 38.0i)T^{2} \) |
| 83 | \( 1 + (-8.80 - 10.6i)T + (-15.5 + 81.5i)T^{2} \) |
| 89 | \( 1 + (4.25 - 11.4i)T + (-67.3 - 58.1i)T^{2} \) |
| 97 | \( 1 + (6.75 + 12.9i)T + (-55.3 + 79.6i)T^{2} \) |
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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
−12.89213330908266481882182072342, −12.36568865366195483464407520499, −11.88565896189864500725612139189, −9.993094584981155264563319027854, −9.653953872355066264227052554898, −7.77079551731493346333591977205, −6.71771395643504926404808254588, −5.71298801262329545751655473494, −4.81012550637428040546829061259, −2.32430883581179637220854239935,
0.43663030608350849946026304685, 3.88031888286736353367426865396, 5.16671941131369083109129622442, 5.94940430680756234948563772780, 6.71593876844970447946840298680, 9.246577983007203631997193422962, 9.909624890693795731176540415364, 10.43939788363618923910117471455, 11.41350319748716029013671475650, 12.80516489224273952374348842254
