| L(s) = 1 | + (−45.2 + 78.3i)2-s + (2.71e3 − 1.56e3i)3-s + (−4.09e3 − 7.09e3i)4-s + (1.01e4 + 5.87e3i)5-s + 2.83e5i·6-s + (5.52e5 + 6.10e5i)7-s + 7.41e5·8-s + (2.50e6 − 4.34e6i)9-s + (−9.20e5 + 5.31e5i)10-s + (7.77e5 + 1.34e6i)11-s + (−2.22e7 − 1.28e7i)12-s − 7.67e7i·13-s + (−7.28e7 + 1.57e7i)14-s + 3.67e7·15-s + (−3.35e7 + 5.81e7i)16-s + (3.82e8 − 2.21e8i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (1.23 − 0.715i)3-s + (−0.249 − 0.433i)4-s + (0.130 + 0.0751i)5-s + 1.01i·6-s + (0.671 + 0.741i)7-s + 0.353·8-s + (0.524 − 0.907i)9-s + (−0.0920 + 0.0531i)10-s + (0.0399 + 0.0691i)11-s + (−0.619 − 0.357i)12-s − 1.22i·13-s + (−0.691 + 0.149i)14-s + 0.215·15-s + (−0.125 + 0.216i)16-s + (0.933 − 0.538i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0855i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.996 - 0.0855i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{15}{2})\) |
\(\approx\) |
\(2.62696 + 0.112548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62696 + 0.112548i\) |
| \(L(8)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (45.2 - 78.3i)T \) |
| 7 | \( 1 + (-5.52e5 - 6.10e5i)T \) |
| good | 3 | \( 1 + (-2.71e3 + 1.56e3i)T + (2.39e6 - 4.14e6i)T^{2} \) |
| 5 | \( 1 + (-1.01e4 - 5.87e3i)T + (3.05e9 + 5.28e9i)T^{2} \) |
| 11 | \( 1 + (-7.77e5 - 1.34e6i)T + (-1.89e14 + 3.28e14i)T^{2} \) |
| 13 | \( 1 + 7.67e7iT - 3.93e15T^{2} \) |
| 17 | \( 1 + (-3.82e8 + 2.21e8i)T + (8.41e16 - 1.45e17i)T^{2} \) |
| 19 | \( 1 + (-1.50e9 - 8.66e8i)T + (3.99e17 + 6.91e17i)T^{2} \) |
| 23 | \( 1 + (4.17e8 - 7.22e8i)T + (-5.79e18 - 1.00e19i)T^{2} \) |
| 29 | \( 1 - 1.93e10T + 2.97e20T^{2} \) |
| 31 | \( 1 + (-3.04e9 + 1.75e9i)T + (3.78e20 - 6.55e20i)T^{2} \) |
| 37 | \( 1 + (1.22e9 - 2.12e9i)T + (-4.50e21 - 7.80e21i)T^{2} \) |
| 41 | \( 1 + 6.79e10iT - 3.79e22T^{2} \) |
| 43 | \( 1 - 8.78e10T + 7.38e22T^{2} \) |
| 47 | \( 1 + (2.20e11 + 1.27e11i)T + (1.28e23 + 2.22e23i)T^{2} \) |
| 53 | \( 1 + (6.20e11 + 1.07e12i)T + (-6.89e23 + 1.19e24i)T^{2} \) |
| 59 | \( 1 + (2.99e12 - 1.72e12i)T + (3.09e24 - 5.36e24i)T^{2} \) |
| 61 | \( 1 + (-3.34e12 - 1.93e12i)T + (4.93e24 + 8.55e24i)T^{2} \) |
| 67 | \( 1 + (4.13e12 + 7.16e12i)T + (-1.83e25 + 3.18e25i)T^{2} \) |
| 71 | \( 1 + 1.12e13T + 8.27e25T^{2} \) |
| 73 | \( 1 + (1.24e13 - 7.21e12i)T + (6.10e25 - 1.05e26i)T^{2} \) |
| 79 | \( 1 + (8.80e12 - 1.52e13i)T + (-1.84e26 - 3.19e26i)T^{2} \) |
| 83 | \( 1 - 5.05e13iT - 7.36e26T^{2} \) |
| 89 | \( 1 + (6.21e13 + 3.59e13i)T + (9.78e26 + 1.69e27i)T^{2} \) |
| 97 | \( 1 - 6.76e13iT - 6.52e27T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.88784812283874352706091605135, −14.61927383593948587339162892621, −13.81500492626159973483953592554, −12.16835251860520448541299083097, −9.841779496724075236796512896754, −8.337060471503626209583159127872, −7.58789406803739463305394377839, −5.54966432186428196926091154675, −2.90957562250479489916797650196, −1.28147778712006646595331550507,
1.41410052147240746022303207320, 3.15054076467162020150596217155, 4.48753573526985722173593470632, 7.62570615637754616159987708758, 8.995052579012893525356894841131, 10.04729233735927797510218178102, 11.58562285359340930929315238037, 13.65342260705334202064859040748, 14.43623480282293575425684697388, 16.07547130633757584054327006354
