| L(s) = 1 | + (−2.82 + i)3-s + i·7-s + (7.00 − 5.65i)9-s − 8.48i·11-s + 15i·13-s + 19.7·17-s − 23·19-s + (−1 − 2.82i)21-s + 2.82·23-s + (−14.1 + 23.0i)27-s − 25.4i·29-s − 33·31-s + (8.48 + 24i)33-s − 66i·37-s + (−15 − 42.4i)39-s + ⋯ |
| L(s) = 1 | + (−0.942 + 0.333i)3-s + 0.142i·7-s + (0.777 − 0.628i)9-s − 0.771i·11-s + 1.15i·13-s + 1.16·17-s − 1.21·19-s + (−0.0476 − 0.134i)21-s + 0.122·23-s + (−0.523 + 0.851i)27-s − 0.877i·29-s − 1.06·31-s + (0.257 + 0.727i)33-s − 1.78i·37-s + (−0.384 − 1.08i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.224546618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.224546618\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.82 - i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - iT - 49T^{2} \) |
| 11 | \( 1 + 8.48iT - 121T^{2} \) |
| 13 | \( 1 - 15iT - 169T^{2} \) |
| 17 | \( 1 - 19.7T + 289T^{2} \) |
| 19 | \( 1 + 23T + 361T^{2} \) |
| 23 | \( 1 - 2.82T + 529T^{2} \) |
| 29 | \( 1 + 25.4iT - 841T^{2} \) |
| 31 | \( 1 + 33T + 961T^{2} \) |
| 37 | \( 1 + 66iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 36.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 45.2T + 2.20e3T^{2} \) |
| 53 | \( 1 + 36.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 39T + 3.72e3T^{2} \) |
| 67 | \( 1 - 113iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 25.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 58iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 70T + 6.24e3T^{2} \) |
| 83 | \( 1 - 152.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 90.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 - iT - 9.40e3T^{2} \) |
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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
−9.545809594649363348071344519439, −8.928815226900928325024680413467, −7.84575956575462738142188534112, −6.88073119118536994344719488889, −6.07201868835543187258337993982, −5.43978089865414802344760698393, −4.34125641719378691634402817525, −3.60819347115328228803540934352, −2.03871160626783299013904520453, −0.61444982009574385522822968934,
0.78792819554787384063924859015, 2.00569157241661335682609786435, 3.42496234790667354451918092332, 4.61021291690372075104315352460, 5.39135108581129731622785555574, 6.15698734276339054596388130120, 7.14443362158050225008454403265, 7.69834310986203812889993004997, 8.683831645769772613811126857335, 9.847216243823946963904857750628
