| L(s) = 1 | + (0.909 + 1.08i)2-s + (1.18 − 3.24i)3-s + (−0.347 + 1.96i)4-s + (4.58 − 1.66i)6-s + (−2.44 + 1.41i)8-s + (−6.83 − 5.73i)9-s + (2.33 + 4.04i)11-s + (5.97 + 3.45i)12-s + (−3.75 − 1.36i)16-s + (−1.45 + 1.22i)17-s − 12.6i·18-s + (4.00 + 1.72i)19-s + (−2.25 + 6.20i)22-s + (1.69 + 9.61i)24-s + (−4.69 + 1.71i)25-s + ⋯ |
| L(s) = 1 | + (0.642 + 0.766i)2-s + (0.681 − 1.87i)3-s + (−0.173 + 0.984i)4-s + (1.87 − 0.681i)6-s + (−0.866 + 0.500i)8-s + (−2.27 − 1.91i)9-s + (0.704 + 1.21i)11-s + (1.72 + 0.996i)12-s + (−0.939 − 0.342i)16-s + (−0.352 + 0.296i)17-s − 2.97i·18-s + (0.918 + 0.394i)19-s + (−0.481 + 1.32i)22-s + (0.346 + 1.96i)24-s + (−0.939 + 0.342i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70153 - 0.198380i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70153 - 0.198380i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 19 | \( 1 + (-4.00 - 1.72i)T \) |
| good | 3 | \( 1 + (-1.18 + 3.24i)T + (-2.29 - 1.92i)T^{2} \) |
| 5 | \( 1 + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.33 - 4.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.45 - 1.22i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + (-1.34 + 3.70i)T + (-31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (2.14 + 12.1i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-9.86 - 11.7i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.91 + 9.43i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (7.22 + 2.62i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-0.170 + 0.294i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.36 - 11.9i)T + (-68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.22 + 9.80i)T + (-16.8 + 95.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.14880359193795375547832457354, −12.20590905229838886927646644991, −11.76390382854928949498166392594, −9.375964422101208504678643954668, −8.365753861823423950510282181136, −7.37457657094967253590996987715, −6.82149037340385372951350959510, −5.66210075499383552096713239828, −3.67489148035696175270220736082, −2.05880234418347244395018623740,
2.86695587920878528688999874646, 3.79912227473024341795502571271, 4.85040987597856448870964717988, 5.96451182560450691026781664244, 8.348823173270527279524169261967, 9.352557615455495874950149151719, 9.932572800734602349222443116711, 11.18382566105165015213552586650, 11.48557251552763405025148352676, 13.35692912592296562357809572104
