L(s) = 1 | + (0.218 + 0.218i)2-s + (1.69 − 0.354i)3-s − 1.90i·4-s + (2.16 + 0.561i)5-s + (0.447 + 0.292i)6-s + (0.852 − 0.852i)8-s + (2.74 − 1.20i)9-s + (0.350 + 0.595i)10-s − 0.762i·11-s + (−0.675 − 3.22i)12-s + (2.27 + 2.27i)13-s + (3.86 + 0.184i)15-s − 3.43·16-s + (−3.43 − 3.43i)17-s + (0.862 + 0.337i)18-s + 1.63i·19-s + ⋯ |
L(s) = 1 | + (0.154 + 0.154i)2-s + (0.978 − 0.204i)3-s − 0.952i·4-s + (0.967 + 0.251i)5-s + (0.182 + 0.119i)6-s + (0.301 − 0.301i)8-s + (0.916 − 0.400i)9-s + (0.110 + 0.188i)10-s − 0.229i·11-s + (−0.194 − 0.932i)12-s + (0.629 + 0.629i)13-s + (0.998 + 0.0476i)15-s − 0.859·16-s + (−0.833 − 0.833i)17-s + (0.203 + 0.0796i)18-s + 0.375i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.58312 - 0.800761i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.58312 - 0.800761i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.69 + 0.354i)T \) |
| 5 | \( 1 + (-2.16 - 0.561i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.218 - 0.218i)T + 2iT^{2} \) |
| 11 | \( 1 + 0.762iT - 11T^{2} \) |
| 13 | \( 1 + (-2.27 - 2.27i)T + 13iT^{2} \) |
| 17 | \( 1 + (3.43 + 3.43i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.63iT - 19T^{2} \) |
| 23 | \( 1 + (5.40 - 5.40i)T - 23iT^{2} \) |
| 29 | \( 1 + 4.94T + 29T^{2} \) |
| 31 | \( 1 - 5.92T + 31T^{2} \) |
| 37 | \( 1 + (2.50 - 2.50i)T - 37iT^{2} \) |
| 41 | \( 1 + 4.35iT - 41T^{2} \) |
| 43 | \( 1 + (-2.69 - 2.69i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.03 + 3.03i)T + 47iT^{2} \) |
| 53 | \( 1 + (4.91 - 4.91i)T - 53iT^{2} \) |
| 59 | \( 1 - 7.69T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 + (-0.0345 + 0.0345i)T - 67iT^{2} \) |
| 71 | \( 1 + 12.4iT - 71T^{2} \) |
| 73 | \( 1 + (0.981 + 0.981i)T + 73iT^{2} \) |
| 79 | \( 1 - 4.23iT - 79T^{2} \) |
| 83 | \( 1 + (-5.05 + 5.05i)T - 83iT^{2} \) |
| 89 | \( 1 + 0.907T + 89T^{2} \) |
| 97 | \( 1 + (3.73 - 3.73i)T - 97iT^{2} \) |
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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
−10.07019224697539828360176190238, −9.425253390418055984851565180728, −8.859768419179854282363505731417, −7.62440311875579794280769345569, −6.63775876485592449338353162696, −6.04126288506839431971809373803, −4.91689069292500304315687319837, −3.73430696723427059225896331031, −2.33689609582914547807454740986, −1.46655060711174407561654768760,
1.90136400353404632172224140915, 2.76365891328844487432135204437, 3.91072358720748148209488730629, 4.71580623692874807365698562540, 6.09997646861009463106910376361, 7.07749499658103024355693769392, 8.291386032329862052475924372829, 8.517982661286876381985159084212, 9.536009499311374268266781208865, 10.34388776617050674212726795726