| L(s) = 1 | + (−0.124 − 1.40i)2-s + (0.711 − 1.95i)3-s + (−1.96 + 0.351i)4-s + (3.16 − 0.558i)5-s + (−2.84 − 0.757i)6-s + (0.199 − 0.345i)7-s + (0.741 + 2.72i)8-s + (−1.01 − 0.851i)9-s + (−1.18 − 4.39i)10-s + (−4.85 + 2.80i)11-s + (−0.713 + 4.09i)12-s + (0.216 + 0.596i)13-s + (−0.512 − 0.238i)14-s + (1.16 − 6.58i)15-s + (3.75 − 1.38i)16-s + (−0.750 + 0.630i)17-s + ⋯ |
| L(s) = 1 | + (−0.0882 − 0.996i)2-s + (0.410 − 1.12i)3-s + (−0.984 + 0.175i)4-s + (1.41 − 0.249i)5-s + (−1.16 − 0.309i)6-s + (0.0754 − 0.130i)7-s + (0.262 + 0.965i)8-s + (−0.338 − 0.283i)9-s + (−0.373 − 1.38i)10-s + (−1.46 + 0.845i)11-s + (−0.205 + 1.18i)12-s + (0.0601 + 0.165i)13-s + (−0.136 − 0.0636i)14-s + (0.299 − 1.70i)15-s + (0.938 − 0.346i)16-s + (−0.182 + 0.152i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.495 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.642395 - 1.10660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.642395 - 1.10660i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.124 + 1.40i)T \) |
| 19 | \( 1 + (2.44 - 3.61i)T \) |
| good | 3 | \( 1 + (-0.711 + 1.95i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-3.16 + 0.558i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.199 + 0.345i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (4.85 - 2.80i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.216 - 0.596i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.750 - 0.630i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.335 - 1.90i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.55 + 4.23i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.06 + 7.04i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.10iT - 37T^{2} \) |
| 41 | \( 1 + (10.4 + 3.79i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.02 + 0.533i)T + (40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (3.18 + 2.67i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-5.58 - 0.984i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (1.02 + 1.22i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.85 + 1.73i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (0.841 - 1.00i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.206 + 1.17i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (7.95 + 2.89i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-7.01 - 2.55i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (11.8 + 6.83i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (1.01 - 0.370i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.60 + 2.18i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.84387642782756000243821769645, −11.95611141490999136786976565010, −10.34047961856698366498485612846, −9.917771127570921234384225469692, −8.538602851431638771664430059116, −7.64340801723708114955444998048, −6.07749149435110478556065622821, −4.74707657544474773682253440327, −2.53443048486037590428698414478, −1.70968650040016289647647593081,
2.96364790899881241154967533271, 4.77506254860228901767974116670, 5.60504773997344673285709124203, 6.79033468004593029844371486554, 8.430199131859644662020050680305, 9.097963578257411802401826533667, 10.24415505481905079711399666897, 10.57986236572777323942046102345, 12.85082863698559893076474026903, 13.66314147554078538084301584100
