| L(s) = 1 | + (33.0 + 33.0i)3-s + (−341. − 341. i)5-s − 1.35e3·7-s + 2.18e3i·9-s + (−7.12e3 + 7.12e3i)11-s + (2.39e4 − 2.39e4i)13-s − 2.25e4i·15-s + 1.42e5·17-s + (1.68e4 + 1.68e4i)19-s + (−4.46e4 − 4.46e4i)21-s − 2.84e5·23-s − 1.57e5i·25-s + (−7.23e4 + 7.23e4i)27-s + (−4.12e4 + 4.12e4i)29-s − 4.92e5i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.546 − 0.546i)5-s − 0.562·7-s + 0.333i·9-s + (−0.486 + 0.486i)11-s + (0.837 − 0.837i)13-s − 0.446i·15-s + 1.70·17-s + (0.129 + 0.129i)19-s + (−0.229 − 0.229i)21-s − 1.01·23-s − 0.402i·25-s + (−0.136 + 0.136i)27-s + (−0.0583 + 0.0583i)29-s − 0.532i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.6355753995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6355753995\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-33.0 - 33.0i)T \) |
| good | 5 | \( 1 + (341. + 341. i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + 1.35e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + (7.12e3 - 7.12e3i)T - 2.14e8iT^{2} \) |
| 13 | \( 1 + (-2.39e4 + 2.39e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 - 1.42e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + (-1.68e4 - 1.68e4i)T + 1.69e10iT^{2} \) |
| 23 | \( 1 + 2.84e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + (4.12e4 - 4.12e4i)T - 5.00e11iT^{2} \) |
| 31 | \( 1 + 4.92e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + (-1.44e5 - 1.44e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 3.50e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + (4.26e6 - 4.26e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 - 2.41e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + (7.04e6 + 7.04e6i)T + 6.22e13iT^{2} \) |
| 59 | \( 1 + (-4.76e6 + 4.76e6i)T - 1.46e14iT^{2} \) |
| 61 | \( 1 + (1.12e7 - 1.12e7i)T - 1.91e14iT^{2} \) |
| 67 | \( 1 + (-1.80e7 - 1.80e7i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 + 5.35e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + 5.24e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 1.76e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (2.65e7 + 2.65e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 - 5.13e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.36e8T + 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.49478205828271910365424137624, −10.18565802934637335886078629871, −9.670901387044422467568663054709, −8.206074207212770507051092340001, −7.85448213546958442471344931226, −6.18843289902980372707211495335, −5.05764477816170434535018434683, −3.84349482271841774389299571680, −2.93155438466535255312642284595, −1.21546899545321111088882879246,
0.14677665735248901061630841373, 1.59624196471688017886032098758, 3.11172739572510215766815527203, 3.76705782803500813765913080264, 5.57028810294473059417485329278, 6.65912720265802935325543536267, 7.61385561180475804118096903561, 8.504536558797599113722438065576, 9.655322198250930225826060612853, 10.68248574739034744396674363398
