L(s) = 1 | + (−0.841 − 2.87i)3-s + (5.98 − 10.3i)7-s + (−7.58 + 4.84i)9-s + (7.43 + 4.29i)11-s + (8.35 + 14.4i)13-s − 22.1i·17-s + 26.9·19-s + (−34.9 − 8.51i)21-s + (33.6 − 19.3i)23-s + (20.3 + 17.7i)27-s + (8.44 + 4.87i)29-s + (6.01 + 10.4i)31-s + (6.10 − 25.0i)33-s − 25.0·37-s + (34.6 − 36.2i)39-s + ⋯ |
L(s) = 1 | + (−0.280 − 0.959i)3-s + (0.855 − 1.48i)7-s + (−0.842 + 0.538i)9-s + (0.675 + 0.390i)11-s + (0.642 + 1.11i)13-s − 1.30i·17-s + 1.41·19-s + (−1.66 − 0.405i)21-s + (1.46 − 0.843i)23-s + (0.753 + 0.657i)27-s + (0.291 + 0.168i)29-s + (0.194 + 0.336i)31-s + (0.184 − 0.758i)33-s − 0.677·37-s + (0.887 − 0.928i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.062578753\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.062578753\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.841 + 2.87i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (-5.98 + 10.3i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-7.43 - 4.29i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.35 - 14.4i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + 22.1iT - 289T^{2} \) |
| 19 | \( 1 - 26.9T + 361T^{2} \) |
| 23 | \( 1 + (-33.6 + 19.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-8.44 - 4.87i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-6.01 - 10.4i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 25.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + (-34.8 + 20.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (39.1 - 67.8i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (64.3 + 37.1i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 - 13.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-32.9 + 18.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4.56 - 7.90i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-12.0 - 20.8i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 43.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 7.68T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.3 + 33.4i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (73.0 + 42.1i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 - 61.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-31.3 + 54.2i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.656066932921010829370555980359, −8.743587866391417680420571633633, −7.77625737585630225250806115048, −6.94781923254983065553501343726, −6.73053499393556486223080470020, −5.15669258066911482737992926577, −4.47640047949453791647798801984, −3.15114572168266904686022783754, −1.56902482812828934153358955192, −0.857847872891334090947170684651,
1.28571495694251888313615972498, 2.92196146156226122414084621865, 3.74396555052997493515692698996, 5.15204582087389053353348959648, 5.50629300566296891353591206662, 6.38509510822134935757299754566, 7.932840697034046077253627443127, 8.625019899323045130190328529839, 9.209325834506819920916580690456, 10.13691345394851198031141424068