L(s) = 1 | + 2.55i·2-s − 3-s − 4.50·4-s − 2.15i·5-s − 2.55i·6-s − 0.830i·7-s − 6.39i·8-s + 9-s + 5.50·10-s + 5.63·11-s + 4.50·12-s + 2.11·13-s + 2.11·14-s + 2.15i·15-s + 7.29·16-s − 5.31·17-s + ⋯ |
L(s) = 1 | + 1.80i·2-s − 0.577·3-s − 2.25·4-s − 0.965i·5-s − 1.04i·6-s − 0.313i·7-s − 2.26i·8-s + 0.333·9-s + 1.74·10-s + 1.69·11-s + 1.30·12-s + 0.587·13-s + 0.566·14-s + 0.557i·15-s + 1.82·16-s − 1.28·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.859632 + 0.648318i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.859632 + 0.648318i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 157 | \( 1 + (3.44 - 12.0i)T \) |
good | 2 | \( 1 - 2.55iT - 2T^{2} \) |
| 5 | \( 1 + 2.15iT - 5T^{2} \) |
| 7 | \( 1 + 0.830iT - 7T^{2} \) |
| 11 | \( 1 - 5.63T + 11T^{2} \) |
| 13 | \( 1 - 2.11T + 13T^{2} \) |
| 17 | \( 1 + 5.31T + 17T^{2} \) |
| 19 | \( 1 - 1.13T + 19T^{2} \) |
| 23 | \( 1 - 0.00420iT - 23T^{2} \) |
| 29 | \( 1 + 7.71iT - 29T^{2} \) |
| 31 | \( 1 - 8.08T + 31T^{2} \) |
| 37 | \( 1 - 6.62T + 37T^{2} \) |
| 41 | \( 1 - 11.4iT - 41T^{2} \) |
| 43 | \( 1 + 8.86iT - 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 1.61iT - 53T^{2} \) |
| 59 | \( 1 - 0.544iT - 59T^{2} \) |
| 61 | \( 1 - 0.548iT - 61T^{2} \) |
| 67 | \( 1 + 7.34T + 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 1.25iT - 73T^{2} \) |
| 79 | \( 1 + 2.79iT - 79T^{2} \) |
| 83 | \( 1 + 7.18iT - 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 10.7iT - 97T^{2} \) |
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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
−11.35241146689751196007595806796, −9.879000230785932240643116668685, −9.009453989380727545160793643827, −8.487674861956575070144169838830, −7.35077873904351344751925741973, −6.41282313081442906044760564085, −5.93884304353792723196765407284, −4.54396489119547466609601358194, −4.23744780209410534657857966381, −0.924916597464505989301927166849,
1.25376996919182396320984512838, 2.61585960602330865306793448708, 3.73384945911979228504534402078, 4.59498187230551866712475760830, 6.10791480607592685267931218430, 6.97361683647581366299684695262, 8.728750497215352950333046033665, 9.279431146086552968181491614241, 10.36067653849599900054614846451, 10.96008110862939772380973607582