L(s) = 1 | + (1.22 + 0.707i)2-s + (−2.94 + 0.548i)3-s + (0.999 + 1.73i)4-s + (7.34 + 4.24i)5-s + (−3.99 − 1.41i)6-s + 2.82i·8-s + (8.39 − 3.23i)9-s + (6 + 10.3i)10-s + (−3.89 − 4.56i)12-s + 13-s + (−23.9 − 8.48i)15-s + (−2.00 + 3.46i)16-s + (7.34 − 4.24i)17-s + (12.5 + 1.97i)18-s + (−15.5 + 26.8i)19-s + 16.9i·20-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.983 + 0.182i)3-s + (0.249 + 0.433i)4-s + (1.46 + 0.848i)5-s + (−0.666 − 0.235i)6-s + 0.353i·8-s + (0.933 − 0.359i)9-s + (0.600 + 1.03i)10-s + (−0.324 − 0.380i)12-s + 0.0769·13-s + (−1.59 − 0.565i)15-s + (−0.125 + 0.216i)16-s + (0.432 − 0.249i)17-s + (0.698 + 0.109i)18-s + (−0.815 + 1.41i)19-s + 0.848i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0569 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0569 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44853 + 1.53356i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.44853 + 1.53356i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (2.94 - 0.548i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-7.34 - 4.24i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - T + 169T^{2} \) |
| 17 | \( 1 + (-7.34 + 4.24i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (15.5 - 26.8i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-7.34 - 4.24i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 31T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-36.7 - 21.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-22.0 + 12.7i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-7.34 + 4.24i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-25 + 43.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (32.5 + 56.2i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 59.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (48.5 + 84.0i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-51.5 + 89.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 42.4iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-102. - 59.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 166T + 9.40e3T^{2} \) |
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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
−11.88318122672153808370623812155, −10.70563934647053939003463214185, −10.25900365313094370236946103020, −9.184524708706837981580887116194, −7.51706184667696002081043103160, −6.46371298184735118213760844151, −5.92283787075638278666634787344, −5.02730190410348177012651326411, −3.51838117218728202682893932838, −1.88273280532843108736259733821,
1.03426613932863629190696564223, 2.31840678816484679206191247234, 4.42037936291710449527370337415, 5.27940385486334611038117734650, 6.03788948382803653504891855423, 6.94314755427139456165040085114, 8.666804336529882820467808655112, 9.732134945260997050218725164131, 10.45089181291948434160687650756, 11.41297074492055615406554443539