| L(s) = 1 | + (−1.14 + 0.549i)2-s + (−1.62 + 0.591i)3-s + (−0.247 + 0.310i)4-s + (1.11 + 1.03i)5-s + (1.53 − 1.56i)6-s + (1.67 + 2.04i)7-s + (0.675 − 2.95i)8-s + (2.30 − 1.92i)9-s + (−1.84 − 0.569i)10-s + (4.51 + 3.08i)11-s + (0.219 − 0.650i)12-s + (−0.984 − 0.671i)13-s + (−3.03 − 1.41i)14-s + (−2.43 − 1.02i)15-s + (0.678 + 2.97i)16-s + (6.40 − 0.965i)17-s + ⋯ |
| L(s) = 1 | + (−0.806 + 0.388i)2-s + (−0.939 + 0.341i)3-s + (−0.123 + 0.155i)4-s + (0.499 + 0.463i)5-s + (0.625 − 0.640i)6-s + (0.634 + 0.772i)7-s + (0.238 − 1.04i)8-s + (0.766 − 0.641i)9-s + (−0.583 − 0.179i)10-s + (1.36 + 0.929i)11-s + (0.0632 − 0.187i)12-s + (−0.273 − 0.186i)13-s + (−0.812 − 0.376i)14-s + (−0.628 − 0.265i)15-s + (0.169 + 0.743i)16-s + (1.55 − 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.467 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.399939 + 0.664238i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.399939 + 0.664238i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.62 - 0.591i)T \) |
| 7 | \( 1 + (-1.67 - 2.04i)T \) |
| good | 2 | \( 1 + (1.14 - 0.549i)T + (1.24 - 1.56i)T^{2} \) |
| 5 | \( 1 + (-1.11 - 1.03i)T + (0.373 + 4.98i)T^{2} \) |
| 11 | \( 1 + (-4.51 - 3.08i)T + (4.01 + 10.2i)T^{2} \) |
| 13 | \( 1 + (0.984 + 0.671i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (-6.40 + 0.965i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (1.48 + 2.58i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.07 - 7.83i)T + (-16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (7.37 - 1.11i)T + (27.7 - 8.54i)T^{2} \) |
| 31 | \( 1 + 0.379T + 31T^{2} \) |
| 37 | \( 1 + (1.36 - 3.47i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (4.25 - 1.31i)T + (33.8 - 23.0i)T^{2} \) |
| 43 | \( 1 + (1.67 + 0.516i)T + (35.5 + 24.2i)T^{2} \) |
| 47 | \( 1 + (-7.36 + 3.54i)T + (29.3 - 36.7i)T^{2} \) |
| 53 | \( 1 + (-1.66 - 4.24i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (0.839 + 3.67i)T + (-53.1 + 25.5i)T^{2} \) |
| 61 | \( 1 + (3.06 + 3.84i)T + (-13.5 + 59.4i)T^{2} \) |
| 67 | \( 1 - 3.91T + 67T^{2} \) |
| 71 | \( 1 + (-8.09 + 10.1i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (10.0 - 6.88i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + 4.04T + 79T^{2} \) |
| 83 | \( 1 + (3.17 - 2.16i)T + (30.3 - 77.2i)T^{2} \) |
| 89 | \( 1 + (0.797 + 10.6i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + (-1.06 + 1.84i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.43847953951416158744687322588, −10.22571069285066909357602358452, −9.573107766754708785429536304395, −8.988098261109069754139306626306, −7.59308266436460210342809723270, −6.91921846231470149492436723154, −5.85202187597628019784721953896, −4.86080343888735438848117562030, −3.58797171770864440187693484136, −1.48949451993426936639009052978,
0.871777074082385997494078044716, 1.67145948555213838783341374509, 4.04612396974538836618207495234, 5.23535574368501409828609278839, 5.97820753087933945309397412070, 7.21242510644275551124830384431, 8.246463144539106724099697041255, 9.140247431666162896209179740344, 10.08298567518225742467154266531, 10.76533722210037823520430760341
