L(s) = 1 | + (4.53 − 4.53i)2-s + (−83.7 + 83.7i)3-s + 214. i·4-s + 3.13i·5-s + 759. i·6-s − 1.56e3·7-s + (2.13e3 + 2.13e3i)8-s − 7.47e3i·9-s + (14.2 + 14.2i)10-s + (1.41e3 − 1.41e3i)11-s + (−1.80e4 − 1.80e4i)12-s − 5.35e4i·13-s + (−7.10e3 + 7.10e3i)14-s + (−262. − 262. i)15-s − 3.56e4·16-s + (−4.61e4 + 4.61e4i)17-s + ⋯ |
L(s) = 1 | + (0.283 − 0.283i)2-s + (−1.03 + 1.03i)3-s + 0.839i·4-s + 0.00501i·5-s + 0.586i·6-s − 0.653·7-s + (0.521 + 0.521i)8-s − 1.13i·9-s + (0.00142 + 0.00142i)10-s + (0.0966 − 0.0966i)11-s + (−0.868 − 0.868i)12-s − 1.87i·13-s + (−0.185 + 0.185i)14-s + (−0.00519 − 0.00519i)15-s − 0.544·16-s + (−0.552 + 0.552i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.688 + 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0568446 - 0.132258i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0568446 - 0.132258i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-3.80e5 + 5.95e5i)T \) |
good | 2 | \( 1 + (-4.53 + 4.53i)T - 256iT^{2} \) |
| 3 | \( 1 + (83.7 - 83.7i)T - 6.56e3iT^{2} \) |
| 5 | \( 1 - 3.13iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 1.56e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (-1.41e3 + 1.41e3i)T - 2.14e8iT^{2} \) |
| 13 | \( 1 + 5.35e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (4.61e4 - 4.61e4i)T - 6.97e9iT^{2} \) |
| 19 | \( 1 + (5.23e4 - 5.23e4i)T - 1.69e10iT^{2} \) |
| 23 | \( 1 + 2.15e5T + 7.83e10T^{2} \) |
| 31 | \( 1 + (-9.66e4 + 9.66e4i)T - 8.52e11iT^{2} \) |
| 37 | \( 1 + (-6.39e5 - 6.39e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 + (8.30e5 + 8.30e5i)T + 7.98e12iT^{2} \) |
| 43 | \( 1 + (3.90e6 - 3.90e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (1.21e6 + 1.21e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + 8.26e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 8.93e6T + 1.46e14T^{2} \) |
| 61 | \( 1 + (1.58e7 - 1.58e7i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 + 8.41e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 2.39e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (1.42e7 + 1.42e7i)T + 8.06e14iT^{2} \) |
| 79 | \( 1 + (-1.40e7 + 1.40e7i)T - 1.51e15iT^{2} \) |
| 83 | \( 1 - 1.12e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (2.74e7 - 2.74e7i)T - 3.93e15iT^{2} \) |
| 97 | \( 1 + (3.01e7 + 3.01e7i)T + 7.83e15iT^{2} \) |
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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
−16.18384032875078830600488916043, −15.18604890844124140444870758630, −13.21482690777306057325990243102, −12.25584338157386060996613961541, −10.97324361355294947229561448395, −10.03463239415376462037933252508, −8.185432469362387155547015205461, −6.12242630839201595928071982337, −4.59901337602835574556344090185, −3.20290373713619865983182465791,
0.06479638303635497132922752378, 1.65835689229963666975431333346, 4.78169997214755412514188738991, 6.41942425744887491643928169914, 6.85289023445495161819864686904, 9.314680761386080709277070119321, 10.91466881076506961394659586099, 12.04426686954560133921817055941, 13.28405008767918666132418211419, 14.31260818788135935406601653542