L(s) = 1 | + (1.11 + 1.11i)2-s + (−0.707 − 0.707i)3-s + 0.476i·4-s − 1.57i·6-s + (−2.09 + 1.62i)7-s + (1.69 − 1.69i)8-s + 1.00i·9-s + 5.20·11-s + (0.336 − 0.336i)12-s + (2.22 + 2.22i)13-s + (−4.13 − 0.523i)14-s + 4.72·16-s + (3.11 − 3.11i)17-s + (−1.11 + 1.11i)18-s + 4.13·19-s + ⋯ |
L(s) = 1 | + (0.786 + 0.786i)2-s + (−0.408 − 0.408i)3-s + 0.238i·4-s − 0.642i·6-s + (−0.790 + 0.612i)7-s + (0.599 − 0.599i)8-s + 0.333i·9-s + 1.56·11-s + (0.0971 − 0.0971i)12-s + (0.618 + 0.618i)13-s + (−1.10 − 0.140i)14-s + 1.18·16-s + (0.755 − 0.755i)17-s + (−0.262 + 0.262i)18-s + 0.947·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95601 + 0.528790i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95601 + 0.528790i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.09 - 1.62i)T \) |
good | 2 | \( 1 + (-1.11 - 1.11i)T + 2iT^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 + (-2.22 - 2.22i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.11 + 3.11i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.13T + 19T^{2} \) |
| 23 | \( 1 + (2.97 - 2.97i)T - 23iT^{2} \) |
| 29 | \( 1 - 0.249iT - 29T^{2} \) |
| 31 | \( 1 + 10.4iT - 31T^{2} \) |
| 37 | \( 1 + (1.69 + 1.69i)T + 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (4.39 - 4.39i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.11 - 3.11i)T - 47iT^{2} \) |
| 53 | \( 1 + (4.89 - 4.89i)T - 53iT^{2} \) |
| 59 | \( 1 + 11.8T + 59T^{2} \) |
| 61 | \( 1 - 2.16iT - 61T^{2} \) |
| 67 | \( 1 + (2.50 + 2.50i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.798T + 71T^{2} \) |
| 73 | \( 1 + (-5.94 - 5.94i)T + 73iT^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 + (-4.59 - 4.59i)T + 83iT^{2} \) |
| 89 | \( 1 + 6.29T + 89T^{2} \) |
| 97 | \( 1 + (-12.1 + 12.1i)T - 97iT^{2} \) |
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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
−11.31599848923635734204221104445, −9.713076511635888514602938168875, −9.402615580157841606340699056314, −7.893755460881057123461912253445, −6.93705918722978572003158563956, −6.20391455820337503189734549421, −5.68627518524736044616004907641, −4.42692833674672777234026480824, −3.36067690295775744496756350530, −1.37274345355280984124771853711,
1.36498382464507369335943265748, 3.42655814104360677671186872032, 3.64918529237321706978931810425, 4.87314268804700502040717147213, 5.99647887415530666153221317475, 6.90917766782205843789579193545, 8.170274827064828491550016226818, 9.262308491460868899419532166076, 10.33056707432903255021814442400, 10.72765034137367049023764408238