| L(s) = 1 | − 1.41·2-s + 2.00·4-s + (3.56 − 3.50i)5-s − 3.17i·7-s − 2.82·8-s + (−5.03 + 4.96i)10-s + 15.6i·11-s + 1.33i·13-s + 4.49i·14-s + 4.00·16-s − 23.4·17-s − 23.3·19-s + (7.12 − 7.01i)20-s − 22.1i·22-s − 28.6·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s + (0.712 − 0.701i)5-s − 0.454i·7-s − 0.353·8-s + (−0.503 + 0.496i)10-s + 1.42i·11-s + 0.102i·13-s + 0.321i·14-s + 0.250·16-s − 1.38·17-s − 1.22·19-s + (0.356 − 0.350i)20-s − 1.00i·22-s − 1.24·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.701 - 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3594649206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3594649206\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-3.56 + 3.50i)T \) |
| good | 7 | \( 1 + 3.17iT - 49T^{2} \) |
| 11 | \( 1 - 15.6iT - 121T^{2} \) |
| 13 | \( 1 - 1.33iT - 169T^{2} \) |
| 17 | \( 1 + 23.4T + 289T^{2} \) |
| 19 | \( 1 + 23.3T + 361T^{2} \) |
| 23 | \( 1 + 28.6T + 529T^{2} \) |
| 29 | \( 1 - 33.1iT - 841T^{2} \) |
| 31 | \( 1 - 18.5T + 961T^{2} \) |
| 37 | \( 1 - 3.95iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 60.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 2.83iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 71.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 14.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 72.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 11.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 3.35iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 40.0iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 86.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 90.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 144. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 74.8iT - 9.40e3T^{2} \) |
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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
−10.13411098515994637686185371395, −9.603032063392164653458028617623, −8.725675225210106168608846407648, −8.010092934861869049971862743560, −6.84855167245432390484117460428, −6.30966319922120445311215474951, −4.91386384610666658227436076024, −4.19662372050026269600386717373, −2.35507227423974783122872826148, −1.55191828464483824012875652012,
0.14048877331465481801848233553, 1.97767213597914056541948157793, 2.77697886503230922270707887355, 4.11266284471223625499359913394, 5.76978085233843572816691411342, 6.18613835872597763430627983901, 7.08893362262391313930775026069, 8.365958599828181698461661366021, 8.742759322375842865784886686826, 9.787376547060781517945329638210
