| L(s) = 1 | + (−1.33 + 1.06i)2-s + (1.50 − 0.863i)3-s + (0.204 − 0.894i)4-s + (0.337 + 0.229i)5-s + (−1.08 + 2.75i)6-s + (−2.64 + 0.0547i)7-s + (−0.802 − 1.66i)8-s + (1.51 − 2.59i)9-s + (−0.695 + 0.0521i)10-s + (−0.812 − 5.39i)11-s + (−0.465 − 1.51i)12-s + (−0.625 − 4.15i)13-s + (3.47 − 2.89i)14-s + (0.704 + 0.0542i)15-s + (4.49 + 2.16i)16-s + (−2.55 + 2.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.944 + 0.753i)2-s + (0.866 − 0.498i)3-s + (0.102 − 0.447i)4-s + (0.150 + 0.102i)5-s + (−0.443 + 1.12i)6-s + (−0.999 + 0.0206i)7-s + (−0.283 − 0.588i)8-s + (0.503 − 0.864i)9-s + (−0.219 + 0.0164i)10-s + (−0.245 − 1.62i)11-s + (−0.134 − 0.438i)12-s + (−0.173 − 1.15i)13-s + (0.928 − 0.772i)14-s + (0.182 + 0.0139i)15-s + (1.12 + 0.541i)16-s + (−0.619 + 0.574i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.746996 - 0.375390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.746996 - 0.375390i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.50 + 0.863i)T \) |
| 7 | \( 1 + (2.64 - 0.0547i)T \) |
| good | 2 | \( 1 + (1.33 - 1.06i)T + (0.445 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-0.337 - 0.229i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.812 + 5.39i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (0.625 + 4.15i)T + (-12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (2.55 - 2.37i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.35 - 3.09i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.61 + 5.23i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (2.64 + 2.84i)T + (-2.16 + 28.9i)T^{2} \) |
| 31 | \( 1 - 8.33iT - 31T^{2} \) |
| 37 | \( 1 + (-7.68 - 2.37i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-0.165 + 2.20i)T + (-40.5 - 6.11i)T^{2} \) |
| 43 | \( 1 + (-0.0259 - 0.346i)T + (-42.5 + 6.40i)T^{2} \) |
| 47 | \( 1 + (2.08 + 2.61i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-0.138 - 0.448i)T + (-43.7 + 29.8i)T^{2} \) |
| 59 | \( 1 + (-4.12 - 1.98i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.38 + 0.316i)T + (54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + (-12.6 - 2.88i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-1.70 + 11.2i)T + (-69.7 - 21.5i)T^{2} \) |
| 79 | \( 1 + 8.34T + 79T^{2} \) |
| 83 | \( 1 + (-5.64 - 0.851i)T + (79.3 + 24.4i)T^{2} \) |
| 89 | \( 1 + (4.15 + 10.5i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (-12.7 + 7.34i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.47862815923418915381235578596, −9.870455946378902582827001121114, −8.860958752054570513050657281080, −8.276810708117950987950400082993, −7.57946866196168508855751199222, −6.44218207156141180560372907135, −5.91070948386490663922738661809, −3.64498323299249013970400776094, −2.86885731296159999495459571346, −0.64016015812063144223743262188,
1.85698767104350888160635146824, 2.79188004197202062090572934136, 4.15515615289201711143235496872, 5.34923763331611173885075328804, 7.05094778303704836461394820171, 7.73670114444607967720566413177, 9.278963656093002974378186167078, 9.456929548051753319573756937850, 9.817302139915353190572213420797, 11.08775011852514093188050885615
