L(s) = 1 | + 0.122i·2-s + (−0.937 + 1.45i)3-s + 1.98·4-s + (0.264 + 0.458i)5-s + (−0.178 − 0.114i)6-s + 0.487i·8-s + (−1.24 − 2.73i)9-s + (−0.0560 + 0.0323i)10-s + (3.64 + 2.10i)11-s + (−1.86 + 2.89i)12-s + (1.74 + 1.00i)13-s + (−0.915 − 0.0444i)15-s + 3.91·16-s + (2.19 + 3.79i)17-s + (0.334 − 0.151i)18-s + (−4.54 − 2.62i)19-s + ⋯ |
L(s) = 1 | + 0.0865i·2-s + (−0.541 + 0.840i)3-s + 0.992·4-s + (0.118 + 0.205i)5-s + (−0.0727 − 0.0468i)6-s + 0.172i·8-s + (−0.413 − 0.910i)9-s + (−0.0177 + 0.0102i)10-s + (1.09 + 0.633i)11-s + (−0.537 + 0.834i)12-s + (0.484 + 0.279i)13-s + (−0.236 − 0.0114i)15-s + 0.977·16-s + (0.532 + 0.921i)17-s + (0.0787 − 0.0358i)18-s + (−1.04 − 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21163 + 0.935008i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21163 + 0.935008i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.937 - 1.45i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.122iT - 2T^{2} \) |
| 5 | \( 1 + (-0.264 - 0.458i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.64 - 2.10i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.74 - 1.00i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.54 + 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.43 - 3.13i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (7.27 - 4.20i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.19iT - 31T^{2} \) |
| 37 | \( 1 + (-1.61 + 2.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0994 + 0.172i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.96 - 6.86i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 + (-3.65 + 2.10i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 13.1iT - 61T^{2} \) |
| 67 | \( 1 + 6.58T + 67T^{2} \) |
| 71 | \( 1 + 8.50iT - 71T^{2} \) |
| 73 | \( 1 + (-4.86 + 2.80i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 0.572T + 79T^{2} \) |
| 83 | \( 1 + (5.42 + 9.39i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.43 + 11.1i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.493 - 0.285i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.10114486994123231242759116854, −10.63044783463003724664430190755, −9.663897647753540270147647378967, −8.759208617061139758212584897383, −7.44909088640768016929848336223, −6.36492305953321046519618931579, −5.93266657190433183776340298324, −4.41693884929319407707494381735, −3.47053563898301788791896491065, −1.80471087565728098913132466129,
1.14123166509193816426109444159, 2.39944454021182578651782880135, 3.89504035088653039544115481801, 5.68157286307946314460810994279, 6.15453164752203169235912075270, 7.14625538321469060838335687339, 7.973071457723733667391932932611, 9.035085825721115411291705192471, 10.36360521689955672152180168026, 11.09931589358254409472428691937