| L(s) = 1 | + (4.41 + 2.34i)5-s + 13.6i·7-s + 12.3i·11-s + 17.0i·13-s + 6.89·17-s + 7.24·19-s − 34.7·23-s + (13.9 + 20.7i)25-s + 21.1i·29-s + 38.2·31-s + (−32.1 + 60.4i)35-s + 21.5i·37-s − 36.3i·41-s + 6.23i·43-s + 40.2·47-s + ⋯ |
| L(s) = 1 | + (0.882 + 0.469i)5-s + 1.95i·7-s + 1.11i·11-s + 1.30i·13-s + 0.405·17-s + 0.381·19-s − 1.51·23-s + (0.558 + 0.829i)25-s + 0.728i·29-s + 1.23·31-s + (−0.918 + 1.72i)35-s + 0.582i·37-s − 0.886i·41-s + 0.145i·43-s + 0.855·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.293775360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.293775360\) |
| \(L(2)\) |
|
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 | \( 1 + (-4.41 - 2.34i)T \) |
| good | 7 | \( 1 - 13.6iT - 49T^{2} \) |
| 11 | \( 1 - 12.3iT - 121T^{2} \) |
| 13 | \( 1 - 17.0iT - 169T^{2} \) |
| 17 | \( 1 - 6.89T + 289T^{2} \) |
| 19 | \( 1 - 7.24T + 361T^{2} \) |
| 23 | \( 1 + 34.7T + 529T^{2} \) |
| 29 | \( 1 - 21.1iT - 841T^{2} \) |
| 31 | \( 1 - 38.2T + 961T^{2} \) |
| 37 | \( 1 - 21.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 36.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 6.23iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 40.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 38.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 41.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 15.0T + 3.72e3T^{2} \) |
| 67 | \( 1 + 128. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 104. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 2.11iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 44.0T + 6.24e3T^{2} \) |
| 83 | \( 1 - 55.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 68.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 101. iT - 9.40e3T^{2} \) |
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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
−9.341688212872702453552199754202, −8.644125814061792401887423500300, −7.67562160207044368056407520325, −6.64603555963723480953343314814, −6.13774839589161563946079705719, −5.34042217646076370073087765278, −4.57551437668375885572538932097, −3.21804391508206378893349653582, −2.16910751017547926740051878067, −1.83984433107337903705633117549,
0.60640116098853111664703144823, 1.12527015810996137296118657494, 2.65947114816755415314769083109, 3.68859255214530588927471863901, 4.42531783591077676190979215805, 5.54322390408332456167204892481, 6.06246853741933668715394170756, 7.03867285917107634325300166543, 7.962572506284074932328621006035, 8.314417041526402608140150653485
