L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)4-s − 0.692i·5-s + (0.866 + 0.499i)6-s + (0.309 + 0.178i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.346 + 0.599i)10-s + (−2.54 + 1.46i)11-s − 0.999·12-s − 0.356·14-s + (−0.599 + 0.346i)15-s + (−0.5 − 0.866i)16-s + (3.35 − 5.81i)17-s − 0.999i·18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 − 0.499i)3-s + (0.249 − 0.433i)4-s − 0.309i·5-s + (0.353 + 0.204i)6-s + (0.116 + 0.0674i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.109 + 0.189i)10-s + (−0.767 + 0.443i)11-s − 0.288·12-s − 0.0953·14-s + (−0.154 + 0.0893i)15-s + (−0.125 − 0.216i)16-s + (0.814 − 1.41i)17-s − 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9597752320\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9597752320\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 0.692iT - 5T^{2} \) |
| 7 | \( 1 + (-0.309 - 0.178i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.54 - 1.46i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.35 + 5.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.24 - 3.60i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.19 - 2.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.91 + 6.78i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 2.76iT - 31T^{2} \) |
| 37 | \( 1 + (8.74 - 5.04i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.23 + 2.44i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.29 + 5.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 4.98iT - 47T^{2} \) |
| 53 | \( 1 + 8.88T + 53T^{2} \) |
| 59 | \( 1 + (-1.42 - 0.821i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.24 + 5.62i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-11.7 + 6.76i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.89 + 3.40i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 3.18iT - 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 14.8iT - 83T^{2} \) |
| 89 | \( 1 + (0.343 - 0.198i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.361 - 0.208i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.744098766711815951207882774899, −9.031559615493827136789697637338, −7.77440228321748126515521691014, −7.63019218901978506129612080374, −6.60002624993524338541346141729, −5.39487877042186424490477859612, −5.08111748364101099438320828865, −3.31368625430908797622048308857, −1.99637027455411371084570881718, −0.65402525548346316310225693858,
1.18188503948486696901808433231, 2.84001474178045465692622315449, 3.57485304709450680385434511054, 4.92252138584941924941332952791, 5.74053395644676258407450387385, 6.86954364585225936548986065560, 7.70700703299697832731758518503, 8.574380789287411888887287745502, 9.360036527974240385301111225557, 10.19566707832058191091282603312