| L(s) = 1 | + (0.568 + 0.223i)2-s + (−1.19 − 1.10i)4-s + (−2.30 − 1.57i)5-s + (0.0953 + 2.64i)7-s + (−0.961 − 1.99i)8-s + (−0.962 − 1.41i)10-s + (0.355 + 2.35i)11-s + (−2.96 + 2.36i)13-s + (−0.536 + 1.52i)14-s + (0.141 + 1.89i)16-s + (−2.98 − 0.920i)17-s + (−5.52 + 3.19i)19-s + (1.01 + 4.43i)20-s + (−0.324 + 1.41i)22-s + (−2.41 − 7.82i)23-s + ⋯ |
| L(s) = 1 | + (0.402 + 0.157i)2-s + (−0.596 − 0.553i)4-s + (−1.03 − 0.704i)5-s + (0.0360 + 0.999i)7-s + (−0.339 − 0.705i)8-s + (−0.304 − 0.446i)10-s + (0.107 + 0.710i)11-s + (−0.823 + 0.656i)13-s + (−0.143 + 0.407i)14-s + (0.0354 + 0.473i)16-s + (−0.724 − 0.223i)17-s + (−1.26 + 0.732i)19-s + (0.226 + 0.991i)20-s + (−0.0690 + 0.302i)22-s + (−0.503 − 1.63i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00792129 + 0.0593060i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00792129 + 0.0593060i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.0953 - 2.64i)T \) |
| good | 2 | \( 1 + (-0.568 - 0.223i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (2.30 + 1.57i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (-0.355 - 2.35i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.96 - 2.36i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.98 + 0.920i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (5.52 - 3.19i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.41 + 7.82i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (8.18 - 1.86i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-3.00 - 1.73i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.29 + 4.91i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-2.49 + 1.20i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-8.98 - 4.32i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.52 + 3.87i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-0.503 + 0.542i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-2.77 + 1.89i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (8.06 + 8.69i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (1.63 - 2.82i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.32 - 0.986i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (8.43 - 3.31i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (5.27 + 9.12i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.22 - 7.81i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (11.8 + 1.78i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 + 3.74iT - 97T^{2} \) |
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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
−11.79125436021007331151010582346, −10.66450687262197975940900252529, −9.491200853903338087499482845219, −8.892087406854099123873228561412, −8.031836908513093163104679378941, −6.75161566280556186196134601424, −5.72870680579247750142783959298, −4.51084548037325553814084355476, −4.23446286729220273411691576354, −2.18202446853057146162310991635,
0.03200717281997268029460932388, 2.80594500032750603173821664824, 3.85383485137412120834060576284, 4.43187978333915075096455411744, 5.89753686369829190660332821363, 7.33785859750047108581801275886, 7.72452740198694681413595455081, 8.770720060063786289488561947610, 9.903741441398848559227198344602, 11.14866297213553197884002062209
