| L(s) = 1 | + (−0.893 − 0.448i)2-s + (−1.71 + 0.231i)3-s + (0.597 + 0.802i)4-s + (−0.267 − 0.892i)5-s + (1.63 + 0.563i)6-s + (1.11 + 2.57i)7-s + (−0.173 − 0.984i)8-s + (2.89 − 0.795i)9-s + (−0.161 + 0.917i)10-s + (4.46 − 1.05i)11-s + (−1.21 − 1.23i)12-s + (1.47 + 0.969i)13-s + (0.163 − 2.80i)14-s + (0.665 + 1.46i)15-s + (−0.286 + 0.957i)16-s + (1.42 − 0.518i)17-s + ⋯ |
| L(s) = 1 | + (−0.631 − 0.317i)2-s + (−0.991 + 0.133i)3-s + (0.298 + 0.401i)4-s + (−0.119 − 0.399i)5-s + (0.668 + 0.229i)6-s + (0.420 + 0.974i)7-s + (−0.0613 − 0.348i)8-s + (0.964 − 0.265i)9-s + (−0.0511 + 0.290i)10-s + (1.34 − 0.318i)11-s + (−0.349 − 0.357i)12-s + (0.408 + 0.268i)13-s + (0.0436 − 0.748i)14-s + (0.171 + 0.379i)15-s + (−0.0717 + 0.239i)16-s + (0.345 − 0.125i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.706721 + 0.0114210i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.706721 + 0.0114210i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.893 + 0.448i)T \) |
| 3 | \( 1 + (1.71 - 0.231i)T \) |
| good | 5 | \( 1 + (0.267 + 0.892i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (-1.11 - 2.57i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-4.46 + 1.05i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (-1.47 - 0.969i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (-1.42 + 0.518i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-4.12 - 1.50i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (2.92 - 6.77i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.418 + 7.19i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (6.24 - 0.730i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (0.516 + 0.433i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (8.46 - 4.25i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-3.77 - 4.00i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-5.01 - 0.586i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (1.33 - 2.31i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.89 - 1.16i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (-3.14 + 4.22i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.832 + 14.2i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (0.0492 - 0.279i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.01 + 11.4i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-3.17 - 1.59i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (9.56 + 4.80i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (-1.74 - 9.90i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (4.96 - 16.5i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−12.25686188625362207625278362148, −11.82016498187471380960532433780, −11.13142784300284100315018494617, −9.725526827923773251948147659701, −9.040403904284734993531093721516, −7.78762591463296461260114984529, −6.38658359153804242816625942365, −5.36218192528410021012739294950, −3.81838966245732108840173169052, −1.44494644829390451379095448270,
1.22191347438888706362493032815, 3.98864540282419771970954605726, 5.44169159369049888110797398104, 6.84029094606588091425388920863, 7.24114097473376897909417948495, 8.745953283705771341158733380909, 10.07520090629794252456980481950, 10.80128036638914143954388612312, 11.60035557697014952500343775685, 12.64150641391354428955428214647
