L(s) = 1 | − 3-s + 5-s + (−1.5 + 2.59i)7-s + 9-s + (−2.5 + 4.33i)11-s + (−0.5 − 0.866i)13-s − 15-s + (3.5 + 6.06i)17-s + (2.5 + 4.33i)19-s + (1.5 − 2.59i)21-s + (−0.5 − 0.866i)23-s + 25-s − 27-s + (−1.5 + 2.59i)29-s + (1.5 − 2.59i)31-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + (−0.566 + 0.981i)7-s + 0.333·9-s + (−0.753 + 1.30i)11-s + (−0.138 − 0.240i)13-s − 0.258·15-s + (0.848 + 1.47i)17-s + (0.573 + 0.993i)19-s + (0.327 − 0.566i)21-s + (−0.104 − 0.180i)23-s + 0.200·25-s − 0.192·27-s + (−0.278 + 0.482i)29-s + (0.269 − 0.466i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.205i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8905771780\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8905771780\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (8 - 1.73i)T \) |
good | 7 | \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.5 + 4.33i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (2.5 + 4.33i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.5 - 7.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.5 + 7.79i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + (-3.5 - 6.06i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.834386875631270470147337739247, −7.962521081336394742696425201352, −7.34652946230400708963861224583, −6.39538521703792654825631984145, −5.71746453793740099762452193787, −5.35956540066640591085463062959, −4.33677419859553142863706998429, −3.33657558080948007539995983905, −2.32393136671671121357694060603, −1.50067273237838381261711393050,
0.30606750883466963191871585572, 1.10851823273325541150960932208, 2.72401406548918202179532631168, 3.29975950529957413986784776491, 4.40778082322961837370031961464, 5.27014477016054648434140693288, 5.75776499627003960135777796725, 6.72789934861155483865984621759, 7.21644469889368692083663739895, 7.965222810914602199925083533824