| L(s) = 1 | + (−1.15 − 0.811i)2-s + (−1.71 + 0.216i)3-s + (0.681 + 1.88i)4-s + (0.0372 + 0.102i)5-s + (2.16 + 1.14i)6-s + (−0.984 + 0.173i)7-s + (0.737 − 2.73i)8-s + (2.90 − 0.744i)9-s + (0.0399 − 0.148i)10-s + (1.44 + 0.526i)11-s + (−1.57 − 3.08i)12-s + (−3.59 − 3.01i)13-s + (1.28 + 0.598i)14-s + (−0.0861 − 0.167i)15-s + (−3.07 + 2.56i)16-s + (1.89 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.818 − 0.574i)2-s + (−0.992 + 0.125i)3-s + (0.340 + 0.940i)4-s + (0.0166 + 0.0457i)5-s + (0.884 + 0.467i)6-s + (−0.372 + 0.0656i)7-s + (0.260 − 0.965i)8-s + (0.968 − 0.248i)9-s + (0.0126 − 0.0470i)10-s + (0.436 + 0.158i)11-s + (−0.455 − 0.890i)12-s + (−0.996 − 0.836i)13-s + (0.342 + 0.159i)14-s + (−0.0222 − 0.0433i)15-s + (−0.767 + 0.640i)16-s + (0.459 + 0.265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.620137 - 0.138468i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.620137 - 0.138468i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.15 + 0.811i)T \) |
| 3 | \( 1 + (1.71 - 0.216i)T \) |
| 7 | \( 1 + (0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.0372 - 0.102i)T + (-3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.44 - 0.526i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (3.59 + 3.01i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 1.09i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.71 - 3.30i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.158 + 0.901i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.10 - 6.08i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.99 - 0.352i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.92 + 5.06i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.36 + 7.59i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.354 - 0.973i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.676 - 3.83i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 5.33iT - 53T^{2} \) |
| 59 | \( 1 + (-13.1 + 4.78i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.31 - 7.47i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-4.10 + 4.89i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-4.10 + 7.10i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.00 + 3.47i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.81 - 10.5i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-9.78 + 8.21i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.13 - 1.22i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-13.3 - 4.84i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.46803658726055227492212919248, −9.689383517885800238411144801017, −8.744047743870298376629132051604, −7.72417904897728561211313076380, −6.83796222490424095311053875178, −6.04378574188321133105404148886, −4.76101824238601058210064482812, −3.71456722156427423154064644351, −2.35442843060388605533665358928, −0.76812381187979502161386958615,
0.791389950537510540593941457862, 2.33532374848747394045383179629, 4.38469508277427973871625989991, 5.18292309238998293852124830621, 6.39713315095791585294546290779, 6.71754753530695148479035657537, 7.66940394118328992133843025288, 8.740070930518128093219703146945, 9.695185586154506491530093125671, 10.15536676745147400014059867739
