| L(s) = 1 | + 3.19i·3-s + 3.90·5-s + 2.64i·7-s − 1.23·9-s − 13.2i·11-s − 1.60·13-s + 12.4i·15-s + 8.10·17-s + 4.35i·19-s − 8.46·21-s + 38.2i·23-s − 9.76·25-s + 24.8i·27-s + 51.1·29-s − 13.6i·31-s + ⋯ |
| L(s) = 1 | + 1.06i·3-s + 0.780·5-s + 0.378i·7-s − 0.137·9-s − 1.20i·11-s − 0.123·13-s + 0.832i·15-s + 0.476·17-s + 0.229i·19-s − 0.403·21-s + 1.66i·23-s − 0.390·25-s + 0.919i·27-s + 1.76·29-s − 0.440i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.343074473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.343074473\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 3.19iT - 9T^{2} \) |
| 5 | \( 1 - 3.90T + 25T^{2} \) |
| 7 | \( 1 - 2.64iT - 49T^{2} \) |
| 11 | \( 1 + 13.2iT - 121T^{2} \) |
| 13 | \( 1 + 1.60T + 169T^{2} \) |
| 17 | \( 1 - 8.10T + 289T^{2} \) |
| 23 | \( 1 - 38.2iT - 529T^{2} \) |
| 29 | \( 1 - 51.1T + 841T^{2} \) |
| 31 | \( 1 + 13.6iT - 961T^{2} \) |
| 37 | \( 1 - 35.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 8.06T + 1.68e3T^{2} \) |
| 43 | \( 1 - 16.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 1.59iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 56.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 106. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 35.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 47.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 125. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 34.2T + 5.32e3T^{2} \) |
| 79 | \( 1 + 71.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 149. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 169.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 109.T + 9.40e3T^{2} \) |
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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
−9.833406545465652023850871286597, −9.120611209632369140963197429306, −8.341850129833912380226479041636, −7.32528055899389571837024400630, −5.98652455001460161943037600834, −5.65723160623421515994750540594, −4.60957493152469277673304765596, −3.61340627164247952969201137483, −2.70661208640096377174349952738, −1.22890207363576015190059390010,
0.77295154281632128294813552644, 1.87087413506338688825032174003, 2.65774149230838199001964147677, 4.25390703833291926502708126605, 5.08169043118663976669155758165, 6.38899170057150556412518965441, 6.69523365651672632512816777290, 7.64074416541464721023478311448, 8.308020868149704360626525313136, 9.492123598116118697279973813312
