| L(s) = 1 | + (0.162 + 1.40i)2-s + (−1.05 − 1.37i)3-s + (−1.94 + 0.456i)4-s + (0.550 + 2.32i)5-s + (1.75 − 1.70i)6-s + (3.79 − 2.82i)7-s + (−0.958 − 2.66i)8-s + (−0.771 + 2.89i)9-s + (−3.17 + 1.15i)10-s + (−2.38 + 2.53i)11-s + (2.68 + 2.19i)12-s + (4.46 + 2.24i)13-s + (4.58 + 4.86i)14-s + (2.60 − 3.20i)15-s + (3.58 − 1.77i)16-s + (−0.831 + 2.28i)17-s + ⋯ |
| L(s) = 1 | + (0.114 + 0.993i)2-s + (−0.609 − 0.792i)3-s + (−0.973 + 0.228i)4-s + (0.246 + 1.03i)5-s + (0.717 − 0.696i)6-s + (1.43 − 1.06i)7-s + (−0.338 − 0.940i)8-s + (−0.257 + 0.966i)9-s + (−1.00 + 0.363i)10-s + (−0.720 + 0.763i)11-s + (0.774 + 0.632i)12-s + (1.23 + 0.621i)13-s + (1.22 + 1.30i)14-s + (0.673 − 0.827i)15-s + (0.895 − 0.444i)16-s + (−0.201 + 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.902469 + 0.788350i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.902469 + 0.788350i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.162 - 1.40i)T \) |
| 3 | \( 1 + (1.05 + 1.37i)T \) |
| good | 5 | \( 1 + (-0.550 - 2.32i)T + (-4.46 + 2.24i)T^{2} \) |
| 7 | \( 1 + (-3.79 + 2.82i)T + (2.00 - 6.70i)T^{2} \) |
| 11 | \( 1 + (2.38 - 2.53i)T + (-0.639 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-4.46 - 2.24i)T + (7.76 + 10.4i)T^{2} \) |
| 17 | \( 1 + (0.831 - 2.28i)T + (-13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-2.09 - 5.74i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.667 + 0.897i)T + (-6.59 - 22.0i)T^{2} \) |
| 29 | \( 1 + (1.22 - 1.86i)T + (-11.4 - 26.6i)T^{2} \) |
| 31 | \( 1 + (4.14 - 1.78i)T + (21.2 - 22.5i)T^{2} \) |
| 37 | \( 1 + (-6.45 + 5.41i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (2.53 - 0.147i)T + (40.7 - 4.75i)T^{2} \) |
| 43 | \( 1 + (6.72 - 2.01i)T + (35.9 - 23.6i)T^{2} \) |
| 47 | \( 1 + (-3.79 + 8.78i)T + (-32.2 - 34.1i)T^{2} \) |
| 53 | \( 1 + (-6.41 + 3.70i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.29 - 3.49i)T + (-3.43 + 58.9i)T^{2} \) |
| 61 | \( 1 + (5.83 - 0.682i)T + (59.3 - 14.0i)T^{2} \) |
| 67 | \( 1 + (4.36 + 6.63i)T + (-26.5 + 61.5i)T^{2} \) |
| 71 | \( 1 + (0.225 + 1.27i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-0.400 + 2.26i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.86 + 0.108i)T + (78.4 + 9.17i)T^{2} \) |
| 83 | \( 1 + (-0.794 + 13.6i)T + (-82.4 - 9.63i)T^{2} \) |
| 89 | \( 1 + (2.15 + 0.379i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (7.20 + 1.70i)T + (86.6 + 43.5i)T^{2} \) |
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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
−11.80316894330290584966781584261, −10.78255889373779896536012014285, −10.29110802578914997218879868075, −8.514966736336084519449495547875, −7.65196068692524465717313342361, −7.09784417851332278994219228235, −6.14836637211677259925456112620, −5.10598863397948114957507222698, −3.91031335605584137515610560072, −1.64214365034138522036677667924,
1.06167142872795312175871194504, 2.87098271979498331501975020784, 4.45452925585066545196545359851, 5.28832070864533883514476700729, 5.68890152352690951152829353872, 8.227259260491993776506348883813, 8.802205175106713534375213680758, 9.519968459526738860331563551317, 10.92982411336756590220102745842, 11.23000490224851414721668423005
