| 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
−12.64150641391354428955428214647, −11.60035557697014952500343775685, −10.80128036638914143954388612312, −10.07520090629794252456980481950, −8.745953283705771341158733380909, −7.24114097473376897909417948495, −6.84029094606588091425388920863, −5.44169159369049888110797398104, −3.98864540282419771970954605726, −1.22191347438888706362493032815,
1.44494644829390451379095448270, 3.81838966245732108840173169052, 5.36218192528410021012739294950, 6.38658359153804242816625942365, 7.78762591463296461260114984529, 9.040403904284734993531093721516, 9.725526827923773251948147659701, 11.13142784300284100315018494617, 11.82016498187471380960532433780, 12.25686188625362207625278362148
