L(s) = 1 | + (0.814 + 1.52i)3-s + (2.08 − 2.08i)5-s + 1.14·7-s + (−1.67 + 2.48i)9-s + (−1.67 − 1.67i)11-s + (0.146 − 0.146i)13-s + (4.88 + 1.48i)15-s + 5.59i·17-s + (−1.48 − 1.48i)19-s + (0.933 + 1.75i)21-s + 3.34i·23-s − 3.68i·25-s + (−5.16 − 0.533i)27-s + (−3.51 − 3.51i)29-s − 5.83i·31-s + ⋯ |
L(s) = 1 | + (0.470 + 0.882i)3-s + (0.931 − 0.931i)5-s + 0.433·7-s + (−0.558 + 0.829i)9-s + (−0.504 − 0.504i)11-s + (0.0405 − 0.0405i)13-s + (1.26 + 0.384i)15-s + 1.35i·17-s + (−0.341 − 0.341i)19-s + (0.203 + 0.382i)21-s + 0.698i·23-s − 0.737i·25-s + (−0.994 − 0.102i)27-s + (−0.652 − 0.652i)29-s − 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48146 + 0.290532i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48146 + 0.290532i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.814 - 1.52i)T \) |
good | 5 | \( 1 + (-2.08 + 2.08i)T - 5iT^{2} \) |
| 7 | \( 1 - 1.14T + 7T^{2} \) |
| 11 | \( 1 + (1.67 + 1.67i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.146 + 0.146i)T - 13iT^{2} \) |
| 17 | \( 1 - 5.59iT - 17T^{2} \) |
| 19 | \( 1 + (1.48 + 1.48i)T + 19iT^{2} \) |
| 23 | \( 1 - 3.34iT - 23T^{2} \) |
| 29 | \( 1 + (3.51 + 3.51i)T + 29iT^{2} \) |
| 31 | \( 1 + 5.83iT - 31T^{2} \) |
| 37 | \( 1 + (4.83 + 4.83i)T + 37iT^{2} \) |
| 41 | \( 1 - 0.610T + 41T^{2} \) |
| 43 | \( 1 + (-1.48 + 1.48i)T - 43iT^{2} \) |
| 47 | \( 1 + 6.41T + 47T^{2} \) |
| 53 | \( 1 + (-0.164 + 0.164i)T - 53iT^{2} \) |
| 59 | \( 1 + (-9.05 - 9.05i)T + 59iT^{2} \) |
| 61 | \( 1 + (-4.53 + 4.53i)T - 61iT^{2} \) |
| 67 | \( 1 + (-0.635 - 0.635i)T + 67iT^{2} \) |
| 71 | \( 1 + 6.90iT - 71T^{2} \) |
| 73 | \( 1 - 7.07iT - 73T^{2} \) |
| 79 | \( 1 - 9.83iT - 79T^{2} \) |
| 83 | \( 1 + (-8.09 + 8.09i)T - 83iT^{2} \) |
| 89 | \( 1 - 0.490T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−12.94688863829536138661796333994, −11.41731086233417170731190715426, −10.48226581411052428380829138898, −9.563000686579759382867326853975, −8.717267954591445525192114150218, −7.911722744896963848696530390207, −5.91045992269051419055218086384, −5.12175414256174822775345333346, −3.85580128647303538734194880823, −2.05501230773373667726771941559,
1.95193416007845952453880695989, 3.05385202407856414320334319242, 5.14623032089333313091968579915, 6.49523910383648292229241378425, 7.19870483065974596795582066592, 8.344727168309171998967759837877, 9.513875552127047611516115155639, 10.47555189588022520503263691679, 11.55902574399441068801817905693, 12.63785601041812546310073542764