| L(s) = 1 | + (−2.82 + 2.82i)2-s − 16.0i·4-s + (−33 − 33i)7-s + (45.2 + 45.2i)8-s − 258. i·11-s + (−213 + 213i)13-s + 186.·14-s − 256.·16-s + (−462. + 462. i)17-s + 722i·19-s + (732 + 732i)22-s + (−810. − 810. i)23-s − 1.20e3i·26-s + (−528. + 528. i)28-s − 2.53e3·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.254 − 0.254i)7-s + (0.250 + 0.250i)8-s − 0.644i·11-s + (−0.349 + 0.349i)13-s + 0.254·14-s − 0.250·16-s + (−0.388 + 0.388i)17-s + 0.458i·19-s + (0.322 + 0.322i)22-s + (−0.319 − 0.319i)23-s − 0.349i·26-s + (−0.127 + 0.127i)28-s − 0.560·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.178587929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.178587929\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 - 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (33 + 33i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 258. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (213 - 213i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + (462. - 462. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 722iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (810. + 810. i)T + 6.43e6iT^{2} \) |
| 29 | \( 1 + 2.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-4.46e3 - 4.46e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 + 7.63e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (3.24e3 - 3.24e3i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (9.92e3 - 9.92e3i)T - 2.29e8iT^{2} \) |
| 53 | \( 1 + (-7.93e3 - 7.93e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-1.98e4 - 1.98e4i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 + 7.12e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.50e4 + 4.50e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 - 2.99e3iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (-2.86e4 - 2.86e4i)T + 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.17e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.11e5 - 1.11e5i)T + 8.58e9iT^{2} \) |
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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
−10.27288378553333429088968954256, −9.458016693973577243314791450965, −8.533489922108317759370808254933, −7.74398830214918006086086090670, −6.69059331396441445409866420086, −5.95736876497616870254661011374, −4.76064984296710647143375558451, −3.55844682421249064978045324893, −2.08076860388148764246787728469, −0.68839361787154385243698380845,
0.51149195291790588830123614423, 1.96035553739673211313585737535, 2.94243193411776356378855516456, 4.20948872548610671824892465948, 5.32619469500539990545816938805, 6.63226598078905504642035692826, 7.51829894996249941134863829487, 8.472834218619313246510806930208, 9.474558842435122288941551504463, 9.990051343627050168846285409542
