L(s) = 1 | − 1.41i·2-s + 2.30i·3-s − 2.00·4-s − i·5-s + 3.25·6-s − 1.30·7-s + 2.82i·8-s − 2.30·9-s − 1.41·10-s − 1.87i·11-s − 4.60i·12-s + 3.97i·13-s + 1.84i·14-s + 2.30·15-s + 4.00·16-s − 7.55·17-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + 1.32i·3-s − 1.00·4-s − 0.447i·5-s + 1.32·6-s − 0.492·7-s + 1.00i·8-s − 0.767·9-s − 0.447·10-s − 0.565i·11-s − 1.32i·12-s + 1.10i·13-s + 0.492i·14-s + 0.594·15-s + 1.00·16-s − 1.83·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(-0.166798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.166798i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 - 2.30iT - 3T^{2} \) |
| 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 1.87iT - 11T^{2} \) |
| 13 | \( 1 - 3.97iT - 13T^{2} \) |
| 17 | \( 1 + 7.55T + 17T^{2} \) |
| 19 | \( 1 + 7.14iT - 19T^{2} \) |
| 29 | \( 1 + 1.77iT - 29T^{2} \) |
| 31 | \( 1 + 6.89T + 31T^{2} \) |
| 37 | \( 1 + 4.22iT - 37T^{2} \) |
| 41 | \( 1 - 6.21T + 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 - 10.7iT - 53T^{2} \) |
| 59 | \( 1 + 3.68iT - 59T^{2} \) |
| 61 | \( 1 - 4.21iT - 61T^{2} \) |
| 67 | \( 1 + 3.68iT - 67T^{2} \) |
| 71 | \( 1 + 7.55T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 7.87T + 79T^{2} \) |
| 83 | \( 1 + 6.85iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−9.567481339752624746120129545624, −9.035854910840383625441269217266, −8.758506748257563515875790626647, −7.11058434328143722661447479612, −5.86117919294973271941612745020, −4.68565403428100657265931943923, −4.32763271629793362556590822392, −3.32633978374305509482808488319, −2.11921543510329103688157666575, −0.07629420755211446275628135852,
1.70726411240092054045034487706, 3.17729084390598741343028016003, 4.44216818788644668724091505965, 5.74392213850421534390185154163, 6.42078148682957966853528719249, 7.02981964992150838675377417124, 7.82334826215695770506028144284, 8.386574674252505531070008516160, 9.556180852188026924173123383381, 10.27831235749785682249115221859