| L(s) = 1 | + (−0.358 + 2.03i)2-s + (1.19 − 0.435i)3-s + (−2.13 − 0.776i)4-s + (2.57 − 0.935i)5-s + (0.457 + 2.59i)6-s + (2.17 − 1.50i)7-s + (0.278 − 0.481i)8-s + (−1.05 + 0.885i)9-s + (0.981 + 5.56i)10-s − 5.30·11-s − 2.89·12-s + (−0.214 − 1.21i)13-s + (2.29 + 4.96i)14-s + (2.67 − 2.24i)15-s + (−2.59 − 2.17i)16-s + (−4.42 − 3.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.253 + 1.43i)2-s + (0.691 − 0.251i)3-s + (−1.06 − 0.388i)4-s + (1.14 − 0.418i)5-s + (0.186 + 1.05i)6-s + (0.821 − 0.570i)7-s + (0.0983 − 0.170i)8-s + (−0.351 + 0.295i)9-s + (0.310 + 1.76i)10-s − 1.60·11-s − 0.834·12-s + (−0.0595 − 0.337i)13-s + (0.612 + 1.32i)14-s + (0.689 − 0.578i)15-s + (−0.649 − 0.544i)16-s + (−1.07 − 0.901i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.988398 + 0.797781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.988398 + 0.797781i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.17 + 1.50i)T \) |
| 19 | \( 1 + (-1.98 - 3.87i)T \) |
| good | 2 | \( 1 + (0.358 - 2.03i)T + (-1.87 - 0.684i)T^{2} \) |
| 3 | \( 1 + (-1.19 + 0.435i)T + (2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (-2.57 + 0.935i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 + (0.214 + 1.21i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (4.42 + 3.71i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.994 - 5.64i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.96 - 0.714i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.21 + 3.83i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.92 + 6.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.533 - 3.02i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.27 + 3.58i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.527 + 0.442i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-6.70 - 2.43i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.436 + 0.366i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (0.920 + 5.22i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.98 - 11.2i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-1.10 - 0.923i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (5.59 - 2.03i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (1.30 + 1.09i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-2.46 - 4.26i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.60 - 1.31i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-15.3 + 5.57i)T + (74.3 - 62.3i)T^{2} \) |
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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
−13.59979228592944637183688331285, −13.29568925353566692997229146759, −11.31307282008273825874468359204, −10.00068774701502946706806534267, −8.897195466495376485244764399860, −7.959295570372617412651631796629, −7.35114175848249063384931607639, −5.68395930430540903131188843447, −5.05384418578623338226522362580, −2.36624254844298824137894510941,
2.20434127801101362294961752223, 2.82732694132616517160864287381, 4.75966887373194352546907322391, 6.33046931188817725147556116272, 8.341898501992835609112638793099, 9.073167675843154135735394464607, 10.15255576755542064902108473859, 10.80418878216845790599696266339, 11.83656364266971606076353926111, 13.06904215944073282220427518433
