| L(s) = 1 | + (0.686 + 0.727i)2-s + (−1.68 + 0.388i)3-s + (−0.0581 + 0.998i)4-s + (−3.71 + 0.434i)5-s + (−1.44 − 0.961i)6-s + (−3.25 − 1.63i)7-s + (−0.766 + 0.642i)8-s + (2.69 − 1.31i)9-s + (−2.86 − 2.40i)10-s + (2.08 + 4.83i)11-s + (−0.289 − 1.70i)12-s + (−0.202 − 0.0480i)13-s + (−1.04 − 3.48i)14-s + (6.10 − 2.17i)15-s + (−0.993 − 0.116i)16-s + (−0.290 + 1.64i)17-s + ⋯ |
| L(s) = 1 | + (0.485 + 0.514i)2-s + (−0.974 + 0.224i)3-s + (−0.0290 + 0.499i)4-s + (−1.66 + 0.194i)5-s + (−0.588 − 0.392i)6-s + (−1.22 − 0.617i)7-s + (−0.270 + 0.227i)8-s + (0.899 − 0.436i)9-s + (−0.906 − 0.760i)10-s + (0.628 + 1.45i)11-s + (−0.0835 − 0.492i)12-s + (−0.0562 − 0.0133i)13-s + (−0.278 − 0.931i)14-s + (1.57 − 0.562i)15-s + (−0.248 − 0.0290i)16-s + (−0.0703 + 0.399i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0510i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0510i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00905092 - 0.354561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00905092 - 0.354561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.686 - 0.727i)T \) |
| 3 | \( 1 + (1.68 - 0.388i)T \) |
| good | 5 | \( 1 + (3.71 - 0.434i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (3.25 + 1.63i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (-2.08 - 4.83i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (0.202 + 0.0480i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (0.290 - 1.64i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.286 - 1.62i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (7.81 - 3.92i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (-0.216 + 0.721i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (4.77 + 3.13i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (-5.15 + 1.87i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-3.43 + 3.63i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (0.625 - 0.839i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.54 + 2.98i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (2.43 - 4.21i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.73 - 10.9i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.123 - 2.12i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-1.79 - 5.98i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (1.97 + 1.65i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (4.28 - 3.59i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.96 - 3.14i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (9.81 + 10.3i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-0.396 + 0.333i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.84 - 0.216i)T + (94.3 + 22.3i)T^{2} \) |
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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
−13.07768205963622296244105715090, −12.24923458307211743096079785931, −11.74842718345762088891536175900, −10.48432740601327069257462274974, −9.475514033873683326817741110817, −7.62166467775192985935214291244, −7.08898737502398254451601861293, −5.99238579768982750031702472148, −4.21347112854335364875159638807, −3.86575156234538976710887605616,
0.31988272978699971607451532457, 3.25847412625980890352784494744, 4.35250024462410606124283185948, 5.84036426994346679103653530145, 6.73699818138902261929748084783, 8.179189970638095400606179187636, 9.448207212497677717045141971783, 10.86222048954233765337992825302, 11.55229263213760161594430072112, 12.24820829997505181692179551791
