| L(s) = 1 | + (0.462 − 2.62i)2-s + (1.69 − 0.617i)3-s + (−4.79 − 1.74i)4-s + (−0.102 + 0.0373i)5-s + (−0.835 − 4.73i)6-s + (0.363 + 2.62i)7-s + (−4.13 + 7.15i)8-s + (0.199 − 0.167i)9-s + (0.0505 + 0.286i)10-s + 4.62·11-s − 9.21·12-s + (−0.448 − 2.54i)13-s + (7.04 + 0.259i)14-s + (−0.151 + 0.126i)15-s + (9.06 + 7.60i)16-s + (−0.974 − 0.817i)17-s + ⋯ |
| L(s) = 1 | + (0.327 − 1.85i)2-s + (0.979 − 0.356i)3-s + (−2.39 − 0.872i)4-s + (−0.0458 + 0.0167i)5-s + (−0.341 − 1.93i)6-s + (0.137 + 0.990i)7-s + (−1.46 + 2.53i)8-s + (0.0666 − 0.0559i)9-s + (0.0159 + 0.0906i)10-s + 1.39·11-s − 2.65·12-s + (−0.124 − 0.704i)13-s + (1.88 + 0.0694i)14-s + (−0.0390 + 0.0327i)15-s + (2.26 + 1.90i)16-s + (−0.236 − 0.198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.768 + 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.489832 - 1.35366i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.489832 - 1.35366i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-0.363 - 2.62i)T \) |
| 19 | \( 1 + (4.18 - 1.20i)T \) |
| good | 2 | \( 1 + (-0.462 + 2.62i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.69 + 0.617i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (0.102 - 0.0373i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + (0.448 + 2.54i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.974 + 0.817i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.151 - 0.856i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.579 + 0.211i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-4.32 + 7.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-4.54 + 7.86i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.82 - 10.3i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.690 + 0.579i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (3.31 - 2.78i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-12.3 - 4.49i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (4.23 + 3.55i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.838 + 4.75i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.313 - 1.77i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (5.80 + 4.86i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (5.71 - 2.07i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.71 - 2.27i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.03 + 5.25i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.78 + 1.37i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (2.47 - 0.902i)T + (74.3 - 62.3i)T^{2} \) |
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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
−12.81398002518851849890050700867, −11.83145273681049278154052089598, −11.15224630977060798966124467700, −9.692241471613308809996434360059, −9.029718735778170481133884295147, −8.091763185932049460434662452105, −5.81978687465814590422635672821, −4.20917240466064136849843649739, −2.94693881231492628651591559218, −1.87006120307435809779470089572,
3.81676832155534455989413944670, 4.48113951778815263418724846871, 6.33405149980437471483357998170, 7.06839071322910622484250281285, 8.366736533024723867516895430944, 8.914855458541974056085758202946, 10.02817185416669985294171960964, 11.94067715094984990592568530660, 13.43089738360354211277245131555, 14.02065940569216888116764904326
