L(s) = 1 | + (4.89 + 2.82i)2-s + (15.9 + 27.7i)4-s + (146. − 84.8i)5-s + (−1 + 1.73i)7-s + 181. i·8-s + 960·10-s + (−29.3 − 16.9i)11-s + (1.47e3 + 2.55e3i)13-s + (−9.79 + 5.65i)14-s + (−512. + 886. i)16-s − 4.48e3i·17-s + 5.25e3·19-s + (4.70e3 + 2.71e3i)20-s + (−96 − 166. i)22-s + (8.87e3 − 5.12e3i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (1.17 − 0.678i)5-s + (−0.00291 + 0.00504i)7-s + 0.353i·8-s + 0.959·10-s + (−0.0220 − 0.0127i)11-s + (0.671 + 1.16i)13-s + (−0.00357 + 0.00206i)14-s + (−0.125 + 0.216i)16-s − 0.911i·17-s + 0.766·19-s + (0.587 + 0.339i)20-s + (−0.00901 − 0.0156i)22-s + (0.729 − 0.421i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(3.905191953\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.905191953\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-146. + 84.8i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 7 | \( 1 + (1 - 1.73i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (29.3 + 16.9i)T + (8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-1.47e3 - 2.55e3i)T + (-2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + 4.48e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 5.25e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-8.87e3 + 5.12e3i)T + (7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-1.91e3 - 1.10e3i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.14e4 + 1.98e4i)T + (-4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 - 3.40e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (-1.45e4 + 8.38e3i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-3.20e3 + 5.54e3i)T + (-3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-1.55e5 - 8.99e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.92e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-2.83e5 + 1.63e5i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-3.12e4 + 5.41e4i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.19e5 + 3.79e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 6.82e4iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 7.30e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (1.70e5 - 2.94e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.29e5 - 2.48e5i)T + (1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.86e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-1.40e5 + 2.43e5i)T + (-4.16e11 - 7.21e11i)T^{2} \) |
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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
−11.99434306903762877615806916579, −10.95922794742768957970359184882, −9.496833191588734921759721070073, −8.929946528673196166724919607536, −7.41322192915653128069404378863, −6.25437516295783102369798934076, −5.35409283033718394134411905096, −4.26712257480328775783731850027, −2.58120050985987963737727057501, −1.19628588494882062098064177755,
1.18935927989187356768736107792, 2.52920982396873063295103589816, 3.61673783899774643682870978292, 5.36914134116636138594873073585, 6.04084209406382272590562337587, 7.24426392605560812004831114944, 8.771564764037043188316300595372, 10.09643764756373628572642491148, 10.55927669873970150735252321856, 11.67336574669003454752376285036