L(s) = 1 | + i·2-s − 4-s + (−1.94 − 3.36i)5-s − i·8-s + (3.36 − 1.94i)10-s + (3.41 + 1.97i)11-s + (2.46 + 1.42i)13-s + 16-s + (0.371 + 0.642i)17-s + (1.54 + 0.892i)19-s + (1.94 + 3.36i)20-s + (−1.97 + 3.41i)22-s + (−5.41 + 3.12i)23-s + (−5.07 + 8.78i)25-s + (−1.42 + 2.46i)26-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.870 − 1.50i)5-s − 0.353i·8-s + (1.06 − 0.615i)10-s + (1.03 + 0.594i)11-s + (0.684 + 0.395i)13-s + 0.250·16-s + (0.0899 + 0.155i)17-s + (0.354 + 0.204i)19-s + (0.435 + 0.753i)20-s + (−0.420 + 0.728i)22-s + (−1.12 + 0.651i)23-s + (−1.01 + 1.75i)25-s + (−0.279 + 0.483i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.475414355\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.475414355\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.94 + 3.36i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3.41 - 1.97i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.46 - 1.42i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.371 - 0.642i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.54 - 0.892i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.41 - 3.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 1.44i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.51iT - 31T^{2} \) |
| 37 | \( 1 + (1.50 - 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.24 + 9.08i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.471 - 0.816i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.18T + 47T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 0.0211T + 59T^{2} \) |
| 61 | \( 1 + 2.46iT - 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 1.94iT - 71T^{2} \) |
| 73 | \( 1 + (-4.20 + 2.42i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 - 3.63T + 79T^{2} \) |
| 83 | \( 1 + (-4.02 - 6.98i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (4.63 - 8.02i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-16.2 + 9.40i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.796079584518851068254461753817, −8.072906726668832311335451150445, −7.53510373503724602472158902769, −6.57820513340057552892480686719, −5.75459742681225378449067374733, −4.94928615149995755671562668980, −4.04581927887046061795359599239, −3.80442852635439281905706896223, −1.77563351607136768769200904371, −0.73366551233809028703707592696,
0.840777804779361973876499413009, 2.31752821302550113300675178437, 3.30201880545546331091232867552, 3.67493883919064882058288843610, 4.60882643260953573958583475187, 5.96907391170473857489915096521, 6.51024810815999420037890570410, 7.37986758876034373794224800454, 8.153029983458322394046786858122, 8.820301502945655151366755654668