| L(s) = 1 | + (0.143 + 0.107i)2-s + (−1.16 + 1.28i)3-s + (−0.564 − 1.88i)4-s + (2.63 + 1.73i)5-s + (−0.304 + 0.0593i)6-s + (−0.657 + 2.56i)7-s + (0.243 − 0.668i)8-s + (−0.280 − 2.98i)9-s + (0.193 + 0.532i)10-s + (−0.910 + 1.81i)11-s + (3.07 + 1.47i)12-s + (4.40 + 1.90i)13-s + (−0.368 + 0.298i)14-s + (−5.30 + 1.35i)15-s + (−3.18 + 2.09i)16-s + (−5.40 − 4.53i)17-s + ⋯ |
| L(s) = 1 | + (0.101 + 0.0756i)2-s + (−0.673 + 0.739i)3-s + (−0.282 − 0.942i)4-s + (1.18 + 0.776i)5-s + (−0.124 + 0.0242i)6-s + (−0.248 + 0.968i)7-s + (0.0859 − 0.236i)8-s + (−0.0935 − 0.995i)9-s + (0.0612 + 0.168i)10-s + (−0.274 + 0.546i)11-s + (0.886 + 0.425i)12-s + (1.22 + 0.527i)13-s + (−0.0985 + 0.0796i)14-s + (−1.36 + 0.350i)15-s + (−0.795 + 0.523i)16-s + (−1.31 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0717 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0717 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.852489 + 0.916035i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.852489 + 0.916035i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.16 - 1.28i)T \) |
| 7 | \( 1 + (0.657 - 2.56i)T \) |
| good | 2 | \( 1 + (-0.143 - 0.107i)T + (0.573 + 1.91i)T^{2} \) |
| 5 | \( 1 + (-2.63 - 1.73i)T + (1.98 + 4.59i)T^{2} \) |
| 11 | \( 1 + (0.910 - 1.81i)T + (-6.56 - 8.82i)T^{2} \) |
| 13 | \( 1 + (-4.40 - 1.90i)T + (8.92 + 9.45i)T^{2} \) |
| 17 | \( 1 + (5.40 + 4.53i)T + (2.95 + 16.7i)T^{2} \) |
| 19 | \( 1 + (-1.76 - 2.09i)T + (-3.29 + 18.7i)T^{2} \) |
| 23 | \( 1 + (-4.21 - 3.97i)T + (1.33 + 22.9i)T^{2} \) |
| 29 | \( 1 + (0.353 - 3.02i)T + (-28.2 - 6.68i)T^{2} \) |
| 31 | \( 1 + (1.97 - 8.35i)T + (-27.7 - 13.9i)T^{2} \) |
| 37 | \( 1 + (-0.150 + 0.851i)T + (-34.7 - 12.6i)T^{2} \) |
| 41 | \( 1 + (-6.23 - 8.37i)T + (-11.7 + 39.2i)T^{2} \) |
| 43 | \( 1 + (0.112 + 1.93i)T + (-42.7 + 4.99i)T^{2} \) |
| 47 | \( 1 + (8.53 - 2.02i)T + (42.0 - 21.0i)T^{2} \) |
| 53 | \( 1 + (8.45 + 4.88i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.04 - 1.02i)T + (35.2 - 47.3i)T^{2} \) |
| 61 | \( 1 + (-6.69 - 2.00i)T + (50.9 + 33.5i)T^{2} \) |
| 67 | \( 1 + (-14.9 + 1.74i)T + (65.1 - 15.4i)T^{2} \) |
| 71 | \( 1 + (-0.218 - 0.599i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-2.15 + 5.92i)T + (-55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-8.33 + 11.1i)T + (-22.6 - 75.6i)T^{2} \) |
| 83 | \( 1 + (-4.41 + 5.92i)T + (-23.8 - 79.5i)T^{2} \) |
| 89 | \( 1 + (-0.323 - 0.117i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (5.13 + 7.80i)T + (-38.4 + 89.0i)T^{2} \) |
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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
−11.01428645446633197213378120553, −9.997630281110728070321228322135, −9.412623854984860958052878931433, −8.910748215443307243443132260616, −6.75609374098508396379405701307, −6.34394296899783803716642257928, −5.43085236074347001875095823552, −4.80808583689363693660110175079, −3.18309456915236411556745737321, −1.72994647914178036748784767493,
0.78247980442222898451268087335, 2.30297217516716096616706533463, 3.87526820000000691615300683895, 4.96708749580182428468790771019, 6.02182931106033046474598352935, 6.74105125697875806418146201875, 7.954461869215548212558308536181, 8.586200863986030162493172439479, 9.578458475343231274244431358799, 10.92009787978349835533846396431
