| L(s) = 1 | + (−1.41 + i)3-s − 2.82i·5-s − 2i·7-s + (1.00 − 2.82i)9-s + 2.82·11-s + 2·13-s + (2.82 + 4.00i)15-s − 6i·19-s + (2 + 2.82i)21-s − 5.65·23-s − 3.00·25-s + (1.41 + 5.00i)27-s + 2.82i·29-s − 2i·31-s + (−4.00 + 2.82i)33-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s − 1.26i·5-s − 0.755i·7-s + (0.333 − 0.942i)9-s + 0.852·11-s + 0.554·13-s + (0.730 + 1.03i)15-s − 1.37i·19-s + (0.436 + 0.617i)21-s − 1.17·23-s − 0.600·25-s + (0.272 + 0.962i)27-s + 0.525i·29-s − 0.359i·31-s + (−0.696 + 0.492i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.810150 - 0.419364i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.810150 - 0.419364i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| good | 5 | \( 1 + 2.82iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 5.65iT - 41T^{2} \) |
| 43 | \( 1 - 2iT - 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 - 8.48iT - 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 14iT - 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 16.9iT - 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−12.27201738026071577932410693463, −11.45366989071993164792525158008, −10.48037837199184057914710551073, −9.393267090732438881586112016761, −8.646397469328374314715580275584, −7.09106364364882632065106526459, −5.91425768741935461375287801931, −4.72276536279775185237603763613, −3.92318506110074745217954875731, −0.985538085087791618495440720323,
2.04039006388695043836256396273, 3.77895800730128944835556879749, 5.67367419108852306454251959543, 6.35884595299066625052224785549, 7.33367081792012781629882529507, 8.537528490647230520807566119173, 10.04178528162597381090836648466, 10.81933237328652092861849136719, 11.84335151118755503591638540983, 12.31016927268196984403134205199
