L(s) = 1 | − 2.40i·2-s + i·3-s − 3.78·4-s + i·5-s + 2.40·6-s + (−1.69 − 2.03i)7-s + 4.29i·8-s − 9-s + 2.40·10-s + (2.71 − 1.90i)11-s − 3.78i·12-s + 4.65·13-s + (−4.89 + 4.06i)14-s − 15-s + 2.76·16-s − 0.659·17-s + ⋯ |
L(s) = 1 | − 1.70i·2-s + 0.577i·3-s − 1.89·4-s + 0.447i·5-s + 0.982·6-s + (−0.638 − 0.769i)7-s + 1.51i·8-s − 0.333·9-s + 0.760·10-s + (0.818 − 0.574i)11-s − 1.09i·12-s + 1.28·13-s + (−1.30 + 1.08i)14-s − 0.258·15-s + 0.691·16-s − 0.159·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9829786042\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9829786042\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (1.69 + 2.03i)T \) |
| 11 | \( 1 + (-2.71 + 1.90i)T \) |
good | 2 | \( 1 + 2.40iT - 2T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 17 | \( 1 + 0.659T + 17T^{2} \) |
| 19 | \( 1 - 0.0804T + 19T^{2} \) |
| 23 | \( 1 + 3.80T + 23T^{2} \) |
| 29 | \( 1 + 8.10iT - 29T^{2} \) |
| 31 | \( 1 + 3.95iT - 31T^{2} \) |
| 37 | \( 1 + 5.99T + 37T^{2} \) |
| 41 | \( 1 + 4.39T + 41T^{2} \) |
| 43 | \( 1 + 5.26iT - 43T^{2} \) |
| 47 | \( 1 + 7.38iT - 47T^{2} \) |
| 53 | \( 1 + 3.67T + 53T^{2} \) |
| 59 | \( 1 + 6.62iT - 59T^{2} \) |
| 61 | \( 1 + 14.5T + 61T^{2} \) |
| 67 | \( 1 + 6.70T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 + 6.71T + 73T^{2} \) |
| 79 | \( 1 + 4.24iT - 79T^{2} \) |
| 83 | \( 1 - 2.35T + 83T^{2} \) |
| 89 | \( 1 + 10.1iT - 89T^{2} \) |
| 97 | \( 1 - 4.04iT - 97T^{2} \) |
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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
−9.642131352214773368795138781869, −8.977718967049569686931313153068, −8.085664666840553895323855559690, −6.67322632858200166521431140384, −5.88322751968125853670842324244, −4.39466616206848346044232007161, −3.71558044793375943839904523437, −3.25573457820779554584635450923, −1.86695619788672407727276642950, −0.44604893229305955766823729844,
1.54944442667164568404059444199, 3.37073671995060707314188684359, 4.55000063374298622326168078948, 5.52465885289145684226983204548, 6.26483727134970170352929406704, 6.73267527896575392959437582994, 7.64662842112878632642891759315, 8.653338633018490953508691408456, 8.861886269536568564620021306482, 9.743661655240950878260322056698