L(s) = 1 | + (5.65 − 9.79i)2-s + (−63.9 − 110. i)4-s + (−664. − 383. i)5-s + (1.66e3 + 1.72e3i)7-s − 1.44e3·8-s + (−7.51e3 + 4.33e3i)10-s + (−2.52e3 − 4.37e3i)11-s + 2.82e4i·13-s + (2.63e4 − 6.56e3i)14-s + (−8.19e3 + 1.41e4i)16-s + (1.35e4 − 7.80e3i)17-s + (1.07e5 + 6.21e4i)19-s + 9.81e4i·20-s − 5.71e4·22-s + (−8.97e4 + 1.55e5i)23-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−1.06 − 0.613i)5-s + (0.694 + 0.719i)7-s − 0.353·8-s + (−0.751 + 0.433i)10-s + (−0.172 − 0.298i)11-s + 0.990i·13-s + (0.686 − 0.170i)14-s + (−0.125 + 0.216i)16-s + (0.161 − 0.0934i)17-s + (0.825 + 0.476i)19-s + 0.613i·20-s − 0.243·22-s + (−0.320 + 0.555i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.598 + 0.801i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.994845463\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.994845463\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-5.65 + 9.79i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.66e3 - 1.72e3i)T \) |
good | 5 | \( 1 + (664. + 383. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (2.52e3 + 4.37e3i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 2.82e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.35e4 + 7.80e3i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.07e5 - 6.21e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (8.97e4 - 1.55e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 7.11e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-8.45e5 + 4.88e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-7.04e5 + 1.21e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 3.98e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 2.21e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (6.69e4 + 3.86e4i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-3.09e6 - 5.36e6i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.27e7 + 7.37e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.14e6 + 1.23e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (1.79e6 + 3.10e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 3.51e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-3.33e5 + 1.92e5i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-1.68e7 + 2.91e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 7.92e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (-4.51e7 - 2.60e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 1.44e8iT - 7.83e15T^{2} \) |
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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
−11.86146585058047799442265191227, −11.01700028711978895115605403755, −9.523837206577462377484773664299, −8.544119789585592108190839341341, −7.56645450491362029373061056076, −5.78548241598423866889222469609, −4.68475485037008671499692956705, −3.68405736817615425968009541417, −2.11751661602913133287897639452, −0.75271132941041507690734203573,
0.75934069155233006886964396686, 2.94440317935982436109891208059, 4.08923322958005059066318934394, 5.16063555477267154636099553422, 6.72557005221555201529976846990, 7.66702006502726114195931050308, 8.244773607612474021837278883003, 10.01079416339235507178092758538, 11.07263670203140273460709686072, 11.93535406738998070114971536541