L(s) = 1 | − 1.92e3i·2-s + (7.83e4 − 6.57e4i)3-s − 1.61e6·4-s + 2.77e7·5-s + (−1.26e8 − 1.50e8i)6-s + (7.43e8 + 7.50e7i)7-s − 9.36e8i·8-s + (1.81e9 − 1.03e10i)9-s − 5.34e10i·10-s + 5.61e10i·11-s + (−1.26e11 + 1.05e11i)12-s + 3.86e11i·13-s + (1.44e11 − 1.43e12i)14-s + (2.17e12 − 1.82e12i)15-s − 5.18e12·16-s + 1.33e13·17-s + ⋯ |
L(s) = 1 | − 1.32i·2-s + (0.766 − 0.642i)3-s − 0.768·4-s + 1.27·5-s + (−0.854 − 1.01i)6-s + (0.994 + 0.100i)7-s − 0.308i·8-s + (0.173 − 0.984i)9-s − 1.69i·10-s + 0.652i·11-s + (−0.588 + 0.493i)12-s + 0.777i·13-s + (0.133 − 1.32i)14-s + (0.973 − 0.816i)15-s − 1.17·16-s + 1.60·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(11)\) |
\(\approx\) |
\(4.436443405\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.436443405\) |
\(L(\frac{23}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-7.83e4 + 6.57e4i)T \) |
| 7 | \( 1 + (-7.43e8 - 7.50e7i)T \) |
good | 2 | \( 1 + 1.92e3iT - 2.09e6T^{2} \) |
| 5 | \( 1 - 2.77e7T + 4.76e14T^{2} \) |
| 11 | \( 1 - 5.61e10iT - 7.40e21T^{2} \) |
| 13 | \( 1 - 3.86e11iT - 2.47e23T^{2} \) |
| 17 | \( 1 - 1.33e13T + 6.90e25T^{2} \) |
| 19 | \( 1 - 3.04e13iT - 7.14e26T^{2} \) |
| 23 | \( 1 + 2.80e14iT - 3.94e28T^{2} \) |
| 29 | \( 1 + 2.28e15iT - 5.13e30T^{2} \) |
| 31 | \( 1 + 3.90e15iT - 2.08e31T^{2} \) |
| 37 | \( 1 - 3.61e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 7.43e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.37e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 1.27e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.55e18iT - 1.62e36T^{2} \) |
| 59 | \( 1 + 6.38e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 1.92e18iT - 3.10e37T^{2} \) |
| 67 | \( 1 - 2.86e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.30e19iT - 7.52e38T^{2} \) |
| 73 | \( 1 + 1.86e19iT - 1.34e39T^{2} \) |
| 79 | \( 1 + 1.57e20T + 7.08e39T^{2} \) |
| 83 | \( 1 - 1.76e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 3.24e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 2.37e20iT - 5.27e41T^{2} \) |
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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
−12.70391322074908179589855654881, −11.80577977335197722903845025054, −10.18100176313355168094137410560, −9.379667404642913168865731471226, −7.82515082542635487693799918309, −6.13840568614792929213324724668, −4.23025222655479909149767620685, −2.60624360850438574235854826479, −1.83793937690020937402946676194, −1.11836811726368479469739094773,
1.49967046990245569681273843664, 2.99957388923556746561127367736, 5.02443029950908040983886663893, 5.69184203043985239044485643561, 7.46771998437267465030625581894, 8.462233745890063762798624964927, 9.646413857556784688761190933077, 11.01454606197142674916427711280, 13.44515715330512122802897759506, 14.22523890458530852359010355381