L(s) = 1 | + (−118 − 204. i)7-s + (−601 + 1.04e3i)13-s − 1.43e3·19-s + (1.56e3 + 2.70e3i)25-s + (5.16e3 − 8.94e3i)31-s + 1.65e4·37-s + (1.67e3 + 2.90e3i)43-s + (−1.94e4 + 3.36e4i)49-s + (1.93e4 + 3.34e4i)61-s + (1.77e4 − 3.07e4i)67-s − 1.45e3·73-s + (5.02e4 + 8.70e4i)79-s + 2.83e5·91-s + (6.71e4 + 1.16e5i)97-s + (−7.04e4 + 1.22e5i)103-s + ⋯ |
L(s) = 1 | + (−0.910 − 1.57i)7-s + (−0.986 + 1.70i)13-s − 0.910·19-s + (0.5 + 0.866i)25-s + (0.964 − 1.67i)31-s + 1.98·37-s + (0.138 + 0.239i)43-s + (−1.15 + 2.00i)49-s + (0.664 + 1.15i)61-s + (0.483 − 0.837i)67-s − 0.0318·73-s + (0.906 + 1.57i)79-s + 3.59·91-s + (0.725 + 1.25i)97-s + (−0.654 + 1.13i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.235676747\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.235676747\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (118 + 204. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (601 - 1.04e3i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-5.16e3 + 8.94e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.65e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-1.67e3 - 2.90e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 + 4.18e8T^{2} \) |
| 59 | \( 1 + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.93e4 - 3.34e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.77e4 + 3.07e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.45e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-5.02e4 - 8.70e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.71e4 - 1.16e5i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−10.85885187208129816786565523270, −9.823550753915984954209018973589, −9.328933667304452982086706881025, −7.83435164372245171197097435108, −6.96888846258784016303826747034, −6.30075341311390343684856286239, −4.55504055508093379641205298998, −3.92307873330906800661831968785, −2.41997702603844572458464687998, −0.837831091847721438811273150360,
0.42824042538507544314208966023, 2.42854857665479453080415965391, 3.08460402345986527145801217700, 4.82513099842539090008239884775, 5.78603993763266492056751332092, 6.61674221111862359330870720080, 8.005400455525024020518596532205, 8.771123095272746360928559416210, 9.791992106742207202275830264154, 10.48874855660387792895866526163