L(s) = 1 | + (0.574 − 1.63i)3-s + (−0.5 − 0.866i)5-s + (−2.28 + 3.96i)7-s + (−2.33 − 1.87i)9-s + (−2.29 + 3.97i)11-s + (1.83 + 3.18i)13-s + (−1.70 + 0.319i)15-s − 2.55·17-s − 4.76·19-s + (5.15 + 6.01i)21-s + (−1.28 − 2.22i)23-s + (−0.499 + 0.866i)25-s + (−4.41 + 2.74i)27-s + (−0.956 + 1.65i)29-s + (1.73 + 3.00i)31-s + ⋯ |
L(s) = 1 | + (0.331 − 0.943i)3-s + (−0.223 − 0.387i)5-s + (−0.864 + 1.49i)7-s + (−0.779 − 0.625i)9-s + (−0.692 + 1.19i)11-s + (0.510 + 0.883i)13-s + (−0.439 + 0.0824i)15-s − 0.619·17-s − 1.09·19-s + (1.12 + 1.31i)21-s + (−0.268 − 0.464i)23-s + (−0.0999 + 0.173i)25-s + (−0.849 + 0.528i)27-s + (−0.177 + 0.307i)29-s + (0.311 + 0.539i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.195 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.404900 + 0.493365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.404900 + 0.493365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.574 + 1.63i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
good | 7 | \( 1 + (2.28 - 3.96i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.29 - 3.97i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.83 - 3.18i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 + 4.76T + 19T^{2} \) |
| 23 | \( 1 + (1.28 + 2.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.956 - 1.65i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.73 - 3.00i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 5.46T + 37T^{2} \) |
| 41 | \( 1 + (3.32 + 5.75i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.27 - 5.67i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.01 + 8.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 4.03T + 53T^{2} \) |
| 59 | \( 1 + (-6.13 - 10.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.79 - 8.30i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.66 + 9.82i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 1.67T + 71T^{2} \) |
| 73 | \( 1 - 3.12T + 73T^{2} \) |
| 79 | \( 1 + (6.64 - 11.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.39 + 9.34i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 4.04T + 89T^{2} \) |
| 97 | \( 1 + (8.68 - 15.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.66361118418787225857295690099, −9.478107488534968648197065651637, −8.835613112373411897139323445304, −8.255110504891502397896502046994, −7.02063915814359884152725403525, −6.40400187838591859079906821088, −5.44019344718675456855556216314, −4.16948387454864967929762440367, −2.67752746497369973745296403098, −1.95501076575529036979157308906,
0.29487432835679363550315035846, 2.81166550529556518504483550168, 3.61501325399575269225583603431, 4.34623542779671272488951931987, 5.72142617360462019782125465864, 6.56482741935942377628999714783, 7.79762437875426766207832524159, 8.354723436084422307080501444403, 9.506912144813544633800040426349, 10.33277971745242440517883439191