L(s) = 1 | + (45 − 67.3i)3-s + 224. i·5-s + 1.75e3·7-s + (−2.51e3 − 6.06e3i)9-s + 6.95e3i·11-s − 2.57e4·13-s + (1.51e4 + 1.01e4i)15-s + 7.48e4i·17-s + 1.89e4·19-s + (7.87e4 − 1.17e5i)21-s + 4.70e5i·23-s + 3.40e5·25-s + (−5.21e5 − 1.03e5i)27-s − 4.60e5i·29-s + 3.51e5·31-s + ⋯ |
L(s) = 1 | + (0.555 − 0.831i)3-s + 0.359i·5-s + 0.728·7-s + (−0.382 − 0.923i)9-s + 0.475i·11-s − 0.900·13-s + (0.298 + 0.199i)15-s + 0.896i·17-s + 0.145·19-s + (0.404 − 0.606i)21-s + 1.68i·23-s + 0.870·25-s + (−0.980 − 0.195i)27-s − 0.651i·29-s + 0.380·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(2.079710934\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.079710934\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-45 + 67.3i)T \) |
good | 5 | \( 1 - 224. iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.75e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 6.95e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 2.57e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 7.48e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.89e4T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.70e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.60e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 3.51e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 1.33e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 1.87e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.52e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.08e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.60e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 - 1.37e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 7.53e5T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.26e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.70e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.76e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 2.29e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 4.63e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.26e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.47e8T + 7.83e15T^{2} \) |
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
−11.41374147273439395643806029111, −10.16087794975541997920810609144, −9.115432885666015484659206527134, −7.966808776946808401961197392347, −7.34540796917537938650444444446, −6.23492168679873949409838594243, −4.87415422025947824325835258399, −3.41550919490135299273803049760, −2.20009132927021261601765470668, −1.24576636162076006430012520972,
0.44415850645929573771848680659, 2.11468252608290202986388957272, 3.24743228928982274160706143898, 4.68283541085945721547247226635, 5.17020925726390311216297228226, 6.92565056130325057527748665359, 8.197969956219001378238944684681, 8.821862544453052927987017647861, 9.900250127812971928510953078256, 10.78027315923164073354451235389