L(s) = 1 | + (−1.95 − 0.422i)2-s + 3.86·3-s + (3.64 + 1.65i)4-s + 4.67·5-s + (−7.56 − 1.63i)6-s + (−6.42 − 4.77i)8-s + 5.97·9-s + (−9.14 − 1.97i)10-s + 14.5i·11-s + (14.0 + 6.39i)12-s + 12.7·13-s + 18.1·15-s + (10.5 + 12.0i)16-s − 19.5i·17-s + (−11.6 − 2.52i)18-s + 17.7·19-s + ⋯ |
L(s) = 1 | + (−0.977 − 0.211i)2-s + 1.28·3-s + (0.910 + 0.413i)4-s + 0.935·5-s + (−1.26 − 0.272i)6-s + (−0.802 − 0.596i)8-s + 0.663·9-s + (−0.914 − 0.197i)10-s + 1.32i·11-s + (1.17 + 0.533i)12-s + 0.977·13-s + 1.20·15-s + (0.658 + 0.752i)16-s − 1.14i·17-s + (−0.648 − 0.140i)18-s + 0.932·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 - 0.0746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.06333 + 0.0770705i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.06333 + 0.0770705i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.95 + 0.422i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 3.86T + 9T^{2} \) |
| 5 | \( 1 - 4.67T + 25T^{2} \) |
| 11 | \( 1 - 14.5iT - 121T^{2} \) |
| 13 | \( 1 - 12.7T + 169T^{2} \) |
| 17 | \( 1 + 19.5iT - 289T^{2} \) |
| 19 | \( 1 - 17.7T + 361T^{2} \) |
| 23 | \( 1 - 8.86T + 529T^{2} \) |
| 29 | \( 1 - 35.4iT - 841T^{2} \) |
| 31 | \( 1 - 29.0iT - 961T^{2} \) |
| 37 | \( 1 + 12.2iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 22.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 79.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 42.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 36.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 2.40T + 3.48e3T^{2} \) |
| 61 | \( 1 - 29.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 40.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 22.6T + 5.04e3T^{2} \) |
| 73 | \( 1 - 76.3iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 136.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 49.9T + 6.88e3T^{2} \) |
| 89 | \( 1 + 1.12iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 158. iT - 9.40e3T^{2} \) |
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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
−10.76468417921489225320919060040, −9.828527423439527239722394003025, −9.272974998103289897943084210399, −8.642440527593917516721074014821, −7.50478734824362496846649437297, −6.84195513191464078641063949173, −5.35992555738865483978513021033, −3.55463528115962194972078545856, −2.50284321853804923799417011763, −1.51589869810938864927689893412,
1.28976481229199723696431499122, 2.54094943417472579101495985748, 3.58434203835612060633884194075, 5.75716081995221145019694199493, 6.31467069486345138546566034374, 7.86962897818802692727590143459, 8.325272047527358500539847944188, 9.203069286172961293975571299779, 9.795900352917887906217357378454, 10.83158815608313617959447318627