L(s) = 1 | + (242. − 18.8i)3-s + 4.81e3i·5-s − 670.·7-s + (5.83e4 − 9.12e3i)9-s + 2.33e5i·11-s − 3.07e5·13-s + (9.07e4 + 1.16e6i)15-s + 6.72e5i·17-s − 1.55e6·19-s + (−1.62e5 + 1.26e4i)21-s − 5.57e6i·23-s − 1.34e7·25-s + (1.39e7 − 3.30e6i)27-s + 2.97e7i·29-s − 3.09e7·31-s + ⋯ |
L(s) = 1 | + (0.996 − 0.0775i)3-s + 1.54i·5-s − 0.0398·7-s + (0.987 − 0.154i)9-s + 1.44i·11-s − 0.828·13-s + (0.119 + 1.53i)15-s + 0.473i·17-s − 0.626·19-s + (−0.0397 + 0.00309i)21-s − 0.866i·23-s − 1.37·25-s + (0.973 − 0.230i)27-s + 1.44i·29-s − 1.08·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0775i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.996 + 0.0775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.715235644\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.715235644\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-242. + 18.8i)T \) |
good | 5 | \( 1 - 4.81e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 + 670.T + 2.82e8T^{2} \) |
| 11 | \( 1 - 2.33e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.07e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 6.72e5iT - 2.01e12T^{2} \) |
| 19 | \( 1 + 1.55e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 5.57e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 2.97e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 + 3.09e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 8.56e7T + 4.80e15T^{2} \) |
| 41 | \( 1 - 3.59e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 3.66e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 3.28e7iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 4.59e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 4.88e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 6.12e7T + 7.13e17T^{2} \) |
| 67 | \( 1 + 6.70e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 1.23e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 1.08e9T + 4.29e18T^{2} \) |
| 79 | \( 1 - 1.86e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 1.09e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 5.19e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 1.07e10T + 7.37e19T^{2} \) |
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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
−10.92147393489568962469108832153, −10.14876937507111928704398226845, −9.392070045300885325946210832207, −8.035789744778377786601266614632, −7.13355117650317986624490376361, −6.58101795147130437623267628498, −4.71803553128641788939446997497, −3.56908189253170827078516892934, −2.54219500353257052941683496170, −1.83467352148474514857011694273,
0.28036671028042017912498655810, 1.30267130746465844141741341327, 2.55061966285347703779829270537, 3.82187960124992596012363110338, 4.79268217449974299983473311373, 5.89600759491835282651997173530, 7.53953831555593046328922094208, 8.335438950634197297758227179348, 9.108594623325349083695238121938, 9.788099123918509381707135897720