L(s) = 1 | − 2.96i·5-s − 7.19·13-s + 7.20i·17-s − 3.80·25-s + 1.27i·29-s + 11.3·37-s + 1.41i·41-s + 7·49-s + 7.07i·53-s + 5.39·61-s + 21.3i·65-s + 13.1·73-s + 21.3·85-s − 5.51i·89-s + 8·97-s + ⋯ |
L(s) = 1 | − 1.32i·5-s − 1.99·13-s + 1.74i·17-s − 0.760·25-s + 0.236i·29-s + 1.87·37-s + 0.220i·41-s + 49-s + 0.971i·53-s + 0.690·61-s + 2.64i·65-s + 1.54·73-s + 2.32·85-s − 0.584i·89-s + 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.472664207\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.472664207\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + 2.96iT - 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 7.19T + 13T^{2} \) |
| 17 | \( 1 - 7.20iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 1.27iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 1.41iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 5.51iT - 89T^{2} \) |
| 97 | \( 1 - 8T + 97T^{2} \) |
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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
−8.113060895490504906690242555203, −7.73358818056306280226128377110, −6.79766761851130566685130877812, −5.90512282240284304788229481496, −5.23515749897040817216075076434, −4.54165886810893450161982865652, −3.98413890076781687611828487419, −2.70807755751182065342865537785, −1.83714056424128204172559856913, −0.76418202422969752001674906828,
0.53854489454888031335303204672, 2.45219871092086943389723068281, 2.52852978264933874074830824163, 3.56514181126161561209769695487, 4.63927313223451585412329548999, 5.21406900061405236171211109271, 6.18843029526990823188166000905, 6.99609708791116622804140114827, 7.32700720501960809393534002482, 7.927538879569691493290280400482