L(s) = 1 | + (0.382 + 0.662i)5-s + (1.73 + i)11-s − 0.317i·13-s + (2.77 − 4.80i)17-s + (3.20 − 1.84i)19-s + (−2.74 + 1.58i)23-s + (2.20 − 3.82i)25-s + 6.82i·29-s + (−5.85 − 3.37i)31-s + (0.121 + 0.210i)37-s + 2.74·41-s + 6.82·43-s + (5.99 + 10.3i)47-s + (10.6 + 6.12i)53-s + 1.53i·55-s + ⋯ |
L(s) = 1 | + (0.171 + 0.296i)5-s + (0.522 + 0.301i)11-s − 0.0879i·13-s + (0.672 − 1.16i)17-s + (0.734 − 0.423i)19-s + (−0.572 + 0.330i)23-s + (0.441 − 0.764i)25-s + 1.26i·29-s + (−1.05 − 0.606i)31-s + (0.0199 + 0.0345i)37-s + 0.428·41-s + 1.04·43-s + (0.873 + 1.51i)47-s + (1.45 + 0.840i)53-s + 0.206i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00394i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.894893124\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.894893124\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.382 - 0.662i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.73 - i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 0.317iT - 13T^{2} \) |
| 17 | \( 1 + (-2.77 + 4.80i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.20 + 1.84i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.74 - 1.58i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6.82iT - 29T^{2} \) |
| 31 | \( 1 + (5.85 + 3.37i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.121 - 0.210i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2.74T + 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 + (-5.99 - 10.3i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.6 - 6.12i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.62 + 11.4i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.08 + 1.78i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.24 + 3.88i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.31iT - 71T^{2} \) |
| 73 | \( 1 + (10.2 + 5.92i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.65 - 9.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.32T + 83T^{2} \) |
| 89 | \( 1 + (-0.831 - 1.43i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.8iT - 97T^{2} \) |
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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
−9.403994616358786456888507757228, −8.617995263810872679555955987350, −7.44699652766330398895863858529, −7.14625186928443349505822257819, −6.03159002419617310191924160549, −5.29532138594017534459044290393, −4.31924712611011745653105014829, −3.29059824647593291480727999540, −2.35714739432557598647573127280, −0.957363309942025848511252379754,
1.02293797138251378167843024267, 2.16484007135508798724745149521, 3.53269259647692223059659195889, 4.15233874100828230156291751744, 5.47112934080609858279378011938, 5.87691809391531154890105821605, 6.96246563932972053640645655955, 7.74437127228498752043536699541, 8.604119521544896815314643261139, 9.196762391314680176577492869429