L(s) = 1 | − 5.41i·2-s + (3.03 − 15.2i)3-s + 2.64·4-s − 41.1·5-s + (−82.8 − 16.4i)6-s + 67.8i·7-s − 187. i·8-s + (−224. − 92.8i)9-s + 223. i·10-s − 718.·11-s + (8.01 − 40.3i)12-s − 17.0·13-s + 367.·14-s + (−125. + 629. i)15-s − 932.·16-s + 1.85e3·17-s + ⋯ |
L(s) = 1 | − 0.957i·2-s + (0.194 − 0.980i)3-s + 0.0825·4-s − 0.736·5-s + (−0.939 − 0.186i)6-s + 0.523i·7-s − 1.03i·8-s + (−0.924 − 0.381i)9-s + 0.705i·10-s − 1.79·11-s + (0.0160 − 0.0809i)12-s − 0.0279·13-s + 0.501·14-s + (−0.143 + 0.722i)15-s − 0.910·16-s + 1.55·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.382206 + 0.873004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.382206 + 0.873004i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-3.03 + 15.2i)T \) |
| 23 | \( 1 + (1.49e3 - 2.05e3i)T \) |
good | 2 | \( 1 + 5.41iT - 32T^{2} \) |
| 5 | \( 1 + 41.1T + 3.12e3T^{2} \) |
| 7 | \( 1 - 67.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 718.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 17.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.85e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.13e3iT - 2.47e6T^{2} \) |
| 29 | \( 1 + 1.74e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 3.91e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.22e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 5.01e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 5.50e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.05e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 3.66e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.00e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 3.03e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 4.26e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.28e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.04e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.71e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 6.57e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.07e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.92e4iT - 8.58e9T^{2} \) |
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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
−12.69620028210346589009463769053, −12.03527118658473244037211974460, −11.12536476028562414312455940320, −9.842630448362559882550877399362, −8.138711793752213970843572406024, −7.33002143813543364172679519988, −5.61749678857529279245899218652, −3.32649546369279729745994590300, −2.19133378192585006996658685223, −0.38960941713970809311380473620,
3.01093419074907673163536614040, 4.68665641069440468131000132951, 5.87948570469647000995107139633, 7.78405655536692300019856726491, 8.088003733395986842587135356958, 10.02777714107427381858969343771, 10.85180941122530642125114245380, 12.14152907662444035458618383310, 13.83784116990488647650182673000, 14.76702579425694410932134810411