L(s) = 1 | − 5.47i·2-s − 3·3-s − 14·4-s + 31·5-s + 16.4i·6-s + 54.7i·7-s − 10.9i·8-s − 72·9-s − 169. i·10-s + (11 + 120. i)11-s + 42·12-s − 186. i·13-s + 300·14-s − 93·15-s − 284·16-s + 230. i·17-s + ⋯ |
L(s) = 1 | − 1.36i·2-s − 0.333·3-s − 0.875·4-s + 1.23·5-s + 0.456i·6-s + 1.11i·7-s − 0.171i·8-s − 0.888·9-s − 1.69i·10-s + (0.0909 + 0.995i)11-s + 0.291·12-s − 1.10i·13-s + 1.53·14-s − 0.413·15-s − 1.10·16-s + 0.795i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0909 + 0.995i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0909 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.794643 - 0.725407i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.794643 - 0.725407i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (-11 - 120. i)T \) |
good | 2 | \( 1 + 5.47iT - 16T^{2} \) |
| 3 | \( 1 + 3T + 81T^{2} \) |
| 5 | \( 1 - 31T + 625T^{2} \) |
| 7 | \( 1 - 54.7iT - 2.40e3T^{2} \) |
| 13 | \( 1 + 186. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 230. iT - 8.35e4T^{2} \) |
| 19 | \( 1 + 98.5iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 277T + 2.79e5T^{2} \) |
| 29 | \( 1 + 1.27e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 167T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.06e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 1.20e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 1.70e3T + 4.87e6T^{2} \) |
| 53 | \( 1 - 4.52e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 2.36e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 3.96e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 2.80e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 3.39e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 3.31e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 6.09e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 832. iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 4.67e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 4.24e3T + 8.85e7T^{2} \) |
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
−19.76891243336154225506294972049, −18.23883335823624907732354596626, −17.34444638482209044977961468755, −15.07365855119871938737353357775, −13.16983060155429795253515143690, −12.05413188563647012617656866518, −10.52479981427678601601763710300, −9.204238464977936325510442516002, −5.73105080156715589507556740723, −2.36292077307959260973677058117,
5.53102188592883621196977418424, 6.90260280784055901568593389407, 8.991403689752017235867616356036, 11.05392227188633231096315632129, 13.74945259582314837992793239083, 14.28695245088176938995164735449, 16.56304172213857674847604833284, 16.87079131996468627874928504157, 18.18253336283652801360653841472, 20.26132514032608998620177529834