L(s) = 1 | + (0.247 + 0.429i)2-s + (−1.59 − 0.667i)3-s + (0.877 − 1.51i)4-s + 3.69·5-s + (−0.109 − 0.851i)6-s + 1.86·8-s + (2.10 + 2.13i)9-s + (0.915 + 1.58i)10-s − 0.892·11-s + (−2.41 + 1.84i)12-s + (−0.598 − 1.03i)13-s + (−5.90 − 2.46i)15-s + (−1.29 − 2.23i)16-s + (0.124 + 0.216i)17-s + (−0.393 + 1.43i)18-s + (−1.40 + 2.43i)19-s + ⋯ |
L(s) = 1 | + (0.175 + 0.303i)2-s + (−0.922 − 0.385i)3-s + (0.438 − 0.759i)4-s + 1.65·5-s + (−0.0447 − 0.347i)6-s + 0.658·8-s + (0.703 + 0.711i)9-s + (0.289 + 0.501i)10-s − 0.269·11-s + (−0.697 + 0.531i)12-s + (−0.165 − 0.287i)13-s + (−1.52 − 0.636i)15-s + (−0.323 − 0.559i)16-s + (0.0303 + 0.0525i)17-s + (−0.0926 + 0.338i)18-s + (−0.322 + 0.557i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.842 + 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61048 - 0.470652i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61048 - 0.470652i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.59 + 0.667i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.247 - 0.429i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 + 0.892T + 11T^{2} \) |
| 13 | \( 1 + (0.598 + 1.03i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.124 - 0.216i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.40 - 2.43i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 2.47T + 23T^{2} \) |
| 29 | \( 1 + (-2.07 + 3.58i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.36 - 4.09i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.39 - 4.14i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.98 - 8.64i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (5.08 + 8.81i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.94 + 8.56i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.906 + 1.56i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.40 - 9.35i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.514 - 0.891i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4.94T + 71T^{2} \) |
| 73 | \( 1 + (-0.915 - 1.58i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.899 - 1.55i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.16 - 10.6i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.20 + 2.08i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.52 - 9.56i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.92713020763194105460946606918, −10.05947060887445059832314377158, −9.774129783649272468932200549523, −8.128316925212301194710902641876, −6.83569446459648529640013252406, −6.25548198439747543778653290690, −5.52241065748632341435235930376, −4.81137805539670338530925593083, −2.39395959018400465555481652163, −1.32429596099997915887046726363,
1.73363410706339068500183610619, 3.01926221001700523568651192945, 4.53876241901627713714388415983, 5.44830216517239354056955716876, 6.47286181666161188654074433809, 7.15880774286087749015060989169, 8.738403560704325938482156645128, 9.606926087885443217493660655857, 10.54814040409818948571801056643, 11.00990307405077073243410591759