| L(s) = 1 | + (−0.5 − 4.97i)5-s − 9.94i·7-s − 9.94i·11-s + 19.8i·13-s − 22·17-s + 4·19-s − 20·23-s + (−24.5 + 4.97i)25-s + 39.7i·29-s − 29·31-s + (−49.5 + 4.97i)35-s − 39.7i·41-s + 19.8i·43-s + 58·47-s − 50·49-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.994i)5-s − 1.42i·7-s − 0.904i·11-s + 1.53i·13-s − 1.29·17-s + 0.210·19-s − 0.869·23-s + (−0.979 + 0.198i)25-s + 1.37i·29-s − 0.935·31-s + (−1.41 + 0.142i)35-s − 0.970i·41-s + 0.462i·43-s + 1.23·47-s − 1.02·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0999 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0999 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2313848607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2313848607\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.5 + 4.97i)T \) |
| good | 7 | \( 1 + 9.94iT - 49T^{2} \) |
| 11 | \( 1 + 9.94iT - 121T^{2} \) |
| 13 | \( 1 - 19.8iT - 169T^{2} \) |
| 17 | \( 1 + 22T + 289T^{2} \) |
| 19 | \( 1 - 4T + 361T^{2} \) |
| 23 | \( 1 + 20T + 529T^{2} \) |
| 29 | \( 1 - 39.7iT - 841T^{2} \) |
| 31 | \( 1 + 29T + 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 + 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 19.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58T + 2.20e3T^{2} \) |
| 53 | \( 1 + 31T + 2.80e3T^{2} \) |
| 59 | \( 1 - 39.7iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 44T + 3.72e3T^{2} \) |
| 67 | \( 1 + 19.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 59.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 89.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 - 19T + 6.88e3T^{2} \) |
| 89 | \( 1 - 59.6iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 129. iT - 9.40e3T^{2} \) |
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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
−8.986531882751316602630882399122, −8.508643937876366161937867662059, −7.46072827074724363018109980971, −6.89082890919588987551501987711, −5.98324647216416591507142801750, −4.93585998804747950058982547326, −4.16729897308219768188476270166, −3.67142608662160561154501678571, −2.03304195073336264616727105349, −1.04621764566211910053047186919,
0.06066150446065525044829889174, 2.09921409985741896898227863919, 2.57082731443644731095264274816, 3.61172093325595762933765424036, 4.70075319463972335738021373770, 5.72040290600090215427023308506, 6.18598165294859868770615559754, 7.18843931977704982789713276515, 7.894669895955875854012691266338, 8.632636516939982820767213749588
