L(s) = 1 | + (2 + 3.46i)2-s + (3.31 + 15.2i)3-s + (−7.99 + 13.8i)4-s + (25.8 − 44.7i)5-s + (−46.1 + 41.9i)6-s + (−53.8 − 93.2i)7-s − 63.9·8-s + (−220. + 101. i)9-s + 206.·10-s + (−60.5 − 104. i)11-s + (−237. − 75.8i)12-s + (315. − 545. i)13-s + (215. − 373. i)14-s + (767. + 245. i)15-s + (−128 − 221. i)16-s + 1.06e3·17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.212 + 0.977i)3-s + (−0.249 + 0.433i)4-s + (0.462 − 0.801i)5-s + (−0.523 + 0.475i)6-s + (−0.415 − 0.719i)7-s − 0.353·8-s + (−0.909 + 0.416i)9-s + 0.654·10-s + (−0.150 − 0.261i)11-s + (−0.476 − 0.152i)12-s + (0.517 − 0.895i)13-s + (0.293 − 0.508i)14-s + (0.881 + 0.281i)15-s + (−0.125 − 0.216i)16-s + 0.897·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0798i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(2.319502231\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.319502231\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 - 3.46i)T \) |
| 3 | \( 1 + (-3.31 - 15.2i)T \) |
| 11 | \( 1 + (60.5 + 104. i)T \) |
good | 5 | \( 1 + (-25.8 + 44.7i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (53.8 + 93.2i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 13 | \( 1 + (-315. + 545. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 - 1.06e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 570.T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-1.16e3 + 2.01e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.52e3 + 4.37e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-2.55e3 + 4.42e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 5.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (4.12e3 - 7.15e3i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-3.04e3 - 5.27e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (2.97e3 + 5.15e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 9.34e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-4.77e3 + 8.27e3i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-8.74e3 - 1.51e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (5.42e3 - 9.40e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 1.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.13e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (2.63e4 + 4.56e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (5.45e3 + 9.44e3i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 - 7.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (3.41e4 + 5.90e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−11.59201382746104725899019516958, −10.36649090229422255360549119897, −9.622657168565222277933492984194, −8.583171398454996933296778735954, −7.69053747389695738590422237763, −6.06466630458133939523704828171, −5.24752562271329979530450502788, −4.13976997931118875590084243625, −3.03205759965271307500057356202, −0.66045573588373672043731108359,
1.34428365382705431362510757489, 2.49758319163037722644583337312, 3.42852705725633243454672107722, 5.41693871262188876225593049280, 6.35826775723461178150982482273, 7.28945645832421099076376944806, 8.746705084676817043660864083420, 9.622400543531671500209482053679, 10.80114628193338164373160108321, 11.79245490963031323144477858500