L(s) = 1 | + (−0.279 + 0.711i)2-s + (1.71 − 0.226i)3-s + (1.03 + 0.963i)4-s + (2.80 − 1.35i)5-s + (−0.318 + 1.28i)6-s + (−2.58 − 0.573i)7-s + (−2.35 + 1.13i)8-s + (2.89 − 0.778i)9-s + (0.177 + 2.37i)10-s + (1.31 + 1.65i)11-s + (2.00 + 1.41i)12-s + (1.04 − 2.65i)13-s + (1.12 − 1.67i)14-s + (4.51 − 2.95i)15-s + (0.0625 + 0.834i)16-s + (−3.99 + 3.70i)17-s + ⋯ |
L(s) = 1 | + (−0.197 + 0.503i)2-s + (0.991 − 0.130i)3-s + (0.518 + 0.481i)4-s + (1.25 − 0.604i)5-s + (−0.129 + 0.524i)6-s + (−0.976 − 0.216i)7-s + (−0.831 + 0.400i)8-s + (0.965 − 0.259i)9-s + (0.0562 + 0.750i)10-s + (0.397 + 0.498i)11-s + (0.577 + 0.409i)12-s + (0.289 − 0.737i)13-s + (0.301 − 0.448i)14-s + (1.16 − 0.763i)15-s + (0.0156 + 0.208i)16-s + (−0.968 + 0.898i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.823 - 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.05601 + 0.638711i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.05601 + 0.638711i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.71 + 0.226i)T \) |
| 7 | \( 1 + (2.58 + 0.573i)T \) |
good | 2 | \( 1 + (0.279 - 0.711i)T + (-1.46 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.80 + 1.35i)T + (3.11 - 3.90i)T^{2} \) |
| 11 | \( 1 + (-1.31 - 1.65i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-1.04 + 2.65i)T + (-9.52 - 8.84i)T^{2} \) |
| 17 | \( 1 + (3.99 - 3.70i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.128 + 0.221i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.402 + 1.76i)T + (-20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (9.28 + 2.86i)T + (23.9 + 16.3i)T^{2} \) |
| 31 | \( 1 + (-3.84 + 6.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 0.509i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-5.65 - 3.85i)T + (14.9 + 38.1i)T^{2} \) |
| 43 | \( 1 + (8.60 - 5.86i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (2.86 - 7.30i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (2.28 - 0.703i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (10.4 - 7.13i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-10.7 + 10.0i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (-1.42 + 2.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.403 - 1.76i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (3.54 + 0.535i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (3.41 + 5.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.58 - 6.58i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.71 + 12.0i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (0.395 - 0.685i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09144619951072079316163974052, −9.813113641246105143060453359941, −9.401462353415010188367319450435, −8.475135146863089705715508367883, −7.61950847091343169622922727054, −6.49949150253297959591144442242, −5.98144397750833101133211606754, −4.21278492362524757925948451178, −2.94484696791072011114611873966, −1.85788406453316281800891640677,
1.77097382964925410867411835583, 2.65909975384032834646959717701, 3.59414278338152852726572322042, 5.47403167136699871886100072067, 6.59584863479863608613073984531, 6.97872681481629662120987203028, 8.892119681308659587245055450449, 9.328922697326371106931977460808, 10.00639072154354577619975819939, 10.79066653901606308588099823749