L(s) = 1 | + (0.530 − 1.31i)2-s − 3-s + (−1.43 − 1.39i)4-s + 2.85i·5-s + (−0.530 + 1.31i)6-s − 4.91·7-s + (−2.58 + 1.14i)8-s + 9-s + (3.74 + 1.51i)10-s + 6.48·11-s + (1.43 + 1.39i)12-s − 2.89i·13-s + (−2.60 + 6.44i)14-s − 2.85i·15-s + (0.131 + 3.99i)16-s + 4.14·17-s + ⋯ |
L(s) = 1 | + (0.375 − 0.927i)2-s − 0.577·3-s + (−0.718 − 0.695i)4-s + 1.27i·5-s + (−0.216 + 0.535i)6-s − 1.85·7-s + (−0.914 + 0.405i)8-s + 0.333·9-s + (1.18 + 0.478i)10-s + 1.95·11-s + (0.414 + 0.401i)12-s − 0.802i·13-s + (−0.696 + 1.72i)14-s − 0.737i·15-s + (0.0329 + 0.999i)16-s + 1.00·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.911350 - 0.758523i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.911350 - 0.758523i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.530 + 1.31i)T \) |
| 3 | \( 1 + T \) |
| 67 | \( 1 + (-6.66 - 4.75i)T \) |
good | 5 | \( 1 - 2.85iT - 5T^{2} \) |
| 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 - 6.48T + 11T^{2} \) |
| 13 | \( 1 + 2.89iT - 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 + 4.77iT - 19T^{2} \) |
| 23 | \( 1 + 6.27iT - 23T^{2} \) |
| 29 | \( 1 - 4.52T + 29T^{2} \) |
| 31 | \( 1 + 0.791T + 31T^{2} \) |
| 37 | \( 1 - 2.50T + 37T^{2} \) |
| 41 | \( 1 - 7.22iT - 41T^{2} \) |
| 43 | \( 1 - 4.34T + 43T^{2} \) |
| 47 | \( 1 + 1.23iT - 47T^{2} \) |
| 53 | \( 1 + 0.123iT - 53T^{2} \) |
| 59 | \( 1 + 2.67iT - 59T^{2} \) |
| 61 | \( 1 + 4.75iT - 61T^{2} \) |
| 71 | \( 1 - 15.6iT - 71T^{2} \) |
| 73 | \( 1 + 1.84T + 73T^{2} \) |
| 79 | \( 1 + 0.728T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 - 0.704T + 89T^{2} \) |
| 97 | \( 1 + 9.89iT - 97T^{2} \) |
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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
−10.05599036065261798099793747131, −9.753202960347239437345303251129, −8.753096345294781164852084164069, −6.98182299133070037115631455728, −6.47034868114777786332937906780, −5.85642298556600676417952632743, −4.32443068854722978801408932951, −3.34453813619880266840154395370, −2.76827198399842470554765678810, −0.75408836983968483008820270384,
1.05778993123057833193106108980, 3.60737899029739618795163992019, 4.06817889677871444552258717890, 5.34600901112021882095812986183, 6.15238534711866578464934147801, 6.66496527924757750633258485735, 7.68525991490337128101654567003, 9.066407549266065270268828636893, 9.242524249393144933020862280571, 10.03472380248515688628612776392