L(s) = 1 | + (−28.0 + 28.0i)2-s + (99.2 + 99.2i)3-s − 554. i·4-s + (2.55e3 − 1.80e3i)5-s − 5.57e3·6-s + (2.01e4 − 2.01e4i)7-s + (−1.31e4 − 1.31e4i)8-s + 1.96e4i·9-s + (−2.09e4 + 1.22e5i)10-s + 1.09e5·11-s + (5.50e4 − 5.50e4i)12-s + (2.55e5 + 2.55e5i)13-s + 1.13e6i·14-s + (4.32e5 + 7.38e4i)15-s + 1.30e6·16-s + (−1.32e6 + 1.32e6i)17-s + ⋯ |
L(s) = 1 | + (−0.877 + 0.877i)2-s + (0.408 + 0.408i)3-s − 0.541i·4-s + (0.816 − 0.577i)5-s − 0.716·6-s + (1.19 − 1.19i)7-s + (−0.402 − 0.402i)8-s + 0.333i·9-s + (−0.209 + 1.22i)10-s + 0.678·11-s + (0.221 − 0.221i)12-s + (0.687 + 0.687i)13-s + 2.10i·14-s + (0.569 + 0.0972i)15-s + 1.24·16-s + (−0.934 + 0.934i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.45579 + 0.657067i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45579 + 0.657067i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-99.2 - 99.2i)T \) |
| 5 | \( 1 + (-2.55e3 + 1.80e3i)T \) |
good | 2 | \( 1 + (28.0 - 28.0i)T - 1.02e3iT^{2} \) |
| 7 | \( 1 + (-2.01e4 + 2.01e4i)T - 2.82e8iT^{2} \) |
| 11 | \( 1 - 1.09e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + (-2.55e5 - 2.55e5i)T + 1.37e11iT^{2} \) |
| 17 | \( 1 + (1.32e6 - 1.32e6i)T - 2.01e12iT^{2} \) |
| 19 | \( 1 + 3.95e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (-1.52e6 - 1.52e6i)T + 4.14e13iT^{2} \) |
| 29 | \( 1 - 1.53e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 2.75e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + (1.67e7 - 1.67e7i)T - 4.80e15iT^{2} \) |
| 41 | \( 1 - 1.16e8T + 1.34e16T^{2} \) |
| 43 | \( 1 + (-7.69e7 - 7.69e7i)T + 2.16e16iT^{2} \) |
| 47 | \( 1 + (1.29e8 - 1.29e8i)T - 5.25e16iT^{2} \) |
| 53 | \( 1 + (3.99e8 + 3.99e8i)T + 1.74e17iT^{2} \) |
| 59 | \( 1 + 1.93e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 4.41e7T + 7.13e17T^{2} \) |
| 67 | \( 1 + (5.17e8 - 5.17e8i)T - 1.82e18iT^{2} \) |
| 71 | \( 1 + 1.06e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-1.19e9 - 1.19e9i)T + 4.29e18iT^{2} \) |
| 79 | \( 1 + 1.89e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + (3.98e8 + 3.98e8i)T + 1.55e19iT^{2} \) |
| 89 | \( 1 + 3.69e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (1.18e10 - 1.18e10i)T - 7.37e19iT^{2} \) |
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
−17.29202867501402592939728426249, −16.05885112907422375197693671580, −14.56064053165668952450225283451, −13.37095841148343914044544586269, −10.95152355704231665696810266118, −9.312161188715095334163470823036, −8.348130962106909960221799643655, −6.69830849488196716783471445582, −4.43243277364632726741643832706, −1.25352511218308021766231294261,
1.47194818106467675104949385838, 2.57403641370506010391669246189, 5.87969555777290264454952474182, 8.267661569866868833063894289508, 9.387079260311698285297091247998, 10.93483044943194957123842193462, 12.09522394518128708888976128911, 14.03768612784580500315226754783, 15.09031392074553354130081117601, 17.56875992490619745359912255030