L(s) = 1 | + (−1.68 + 0.418i)3-s + (−1.37 + 2.37i)5-s + (2.65 − 1.40i)9-s + (−0.362 + 0.209i)11-s + (−1.32 + 0.765i)13-s + (1.31 − 4.56i)15-s + (1.95 − 3.38i)17-s + (5.11 − 2.95i)19-s + (7.72 + 4.46i)23-s + (−1.26 − 2.18i)25-s + (−3.86 + 3.47i)27-s + (6.00 + 3.46i)29-s + 3.52i·31-s + (0.521 − 0.503i)33-s + (−4.54 − 7.87i)37-s + ⋯ |
L(s) = 1 | + (−0.970 + 0.241i)3-s + (−0.613 + 1.06i)5-s + (0.883 − 0.468i)9-s + (−0.109 + 0.0630i)11-s + (−0.367 + 0.212i)13-s + (0.338 − 1.17i)15-s + (0.473 − 0.820i)17-s + (1.17 − 0.678i)19-s + (1.61 + 0.930i)23-s + (−0.252 − 0.437i)25-s + (−0.744 + 0.667i)27-s + (1.11 + 0.643i)29-s + 0.633i·31-s + (0.0907 − 0.0875i)33-s + (−0.747 − 1.29i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.247 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.247 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9538100392\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9538100392\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.68 - 0.418i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.37 - 2.37i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.362 - 0.209i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.32 - 0.765i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.95 + 3.38i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.11 + 2.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.72 - 4.46i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.00 - 3.46i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 3.52iT - 31T^{2} \) |
| 37 | \( 1 + (4.54 + 7.87i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.06 + 1.84i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.77 - 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 1.77T + 47T^{2} \) |
| 53 | \( 1 + (3.39 + 1.96i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 4.05T + 59T^{2} \) |
| 61 | \( 1 + 1.86iT - 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 - 8.51iT - 71T^{2} \) |
| 73 | \( 1 + (1.65 + 0.952i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 0.867T + 79T^{2} \) |
| 83 | \( 1 + (3.45 - 5.99i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.88 - 8.46i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.200 + 0.115i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.665714884451154898191145736237, −8.922065229831740735904713316018, −7.44908981034615907035613558278, −7.21870112202400022829665381487, −6.49853049662495101979170743053, −5.26777853158553069725987916456, −4.86504739152790691561047494693, −3.52526084963274087351181790026, −2.90021134572226208254105702661, −1.08254617720935293591800958203,
0.52504434357393251714182874969, 1.49647469956505615893160146634, 3.15992738318111649004703074605, 4.38192385861945943789163403488, 4.96716776166213603875017606385, 5.70831599375658850593648059336, 6.64671324175922362113659119344, 7.53054898120822103115543065312, 8.211845401776610124255022881345, 8.944057402936510579842467703526