| L(s) = 1 | + 1.38·5-s − 1.12·7-s + 2.22·11-s − 16.6i·13-s − 20.2i·17-s − 21.0i·19-s + 20.8i·23-s − 23.0·25-s − 53.9·29-s + 18.3·31-s − 1.56·35-s + 34.8i·37-s − 17.4i·41-s + 51.0i·43-s + 37.4i·47-s + ⋯ |
| L(s) = 1 | + 0.277·5-s − 0.160·7-s + 0.202·11-s − 1.28i·13-s − 1.19i·17-s − 1.10i·19-s + 0.905i·23-s − 0.923·25-s − 1.85·29-s + 0.593·31-s − 0.0445·35-s + 0.941i·37-s − 0.425i·41-s + 1.18i·43-s + 0.797i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2771968840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2771968840\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1.38T + 25T^{2} \) |
| 7 | \( 1 + 1.12T + 49T^{2} \) |
| 11 | \( 1 - 2.22T + 121T^{2} \) |
| 13 | \( 1 + 16.6iT - 169T^{2} \) |
| 17 | \( 1 + 20.2iT - 289T^{2} \) |
| 19 | \( 1 + 21.0iT - 361T^{2} \) |
| 23 | \( 1 - 20.8iT - 529T^{2} \) |
| 29 | \( 1 + 53.9T + 841T^{2} \) |
| 31 | \( 1 - 18.3T + 961T^{2} \) |
| 37 | \( 1 - 34.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 17.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 37.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 52.6T + 2.80e3T^{2} \) |
| 59 | \( 1 - 54.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 88.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 76.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 82.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 39.7T + 6.24e3T^{2} \) |
| 83 | \( 1 + 113.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 25.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 20.8T + 9.40e3T^{2} \) |
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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
−8.240595833824880943947304422608, −7.51802556123682283018231616595, −6.87697287774887895004886488932, −5.81618244195447263303306933327, −5.32616613549858534810397489423, −4.34509767608896547754607605834, −3.26239723581692411248697743181, −2.56001348572352426362957505874, −1.25044816778194020205099057912, −0.06318195738045046569508285611,
1.59837706217937019336434106887, 2.22372582838924788154474817419, 3.76188668895967532154022733908, 4.06757695803117449548271989315, 5.34356837266272808397793298875, 6.07095888257919318586338734382, 6.70836152280878384375650782096, 7.59048850437274051562139882966, 8.418244372640813135289412226733, 9.079562131467886094686014030341
