L(s) = 1 | + (21.8 − 23.3i)2-s − 416.·3-s + (−71.0 − 1.02e3i)4-s + (−2.53e3 − 1.83e3i)5-s + (−9.09e3 + 9.75e3i)6-s + 9.55e3·7-s + (−2.54e4 − 2.06e4i)8-s + 1.14e5·9-s + (−9.81e4 + 1.92e4i)10-s + 3.04e5i·11-s + (2.96e4 + 4.25e5i)12-s − 2.93e4i·13-s + (2.08e5 − 2.23e5i)14-s + (1.05e6 + 7.63e5i)15-s + (−1.03e6 + 1.45e5i)16-s − 1.45e6i·17-s + ⋯ |
L(s) = 1 | + (0.682 − 0.731i)2-s − 1.71·3-s + (−0.0694 − 0.997i)4-s + (−0.810 − 0.585i)5-s + (−1.16 + 1.25i)6-s + 0.568·7-s + (−0.776 − 0.629i)8-s + 1.94·9-s + (−0.981 + 0.192i)10-s + 1.89i·11-s + (0.119 + 1.71i)12-s − 0.0791i·13-s + (0.387 − 0.415i)14-s + (1.38 + 1.00i)15-s + (−0.990 + 0.138i)16-s − 1.02i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.528 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(0.362417 + 0.201337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.362417 + 0.201337i\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-21.8 + 23.3i)T \) |
| 5 | \( 1 + (2.53e3 + 1.83e3i)T \) |
good | 3 | \( 1 + 416.T + 5.90e4T^{2} \) |
| 7 | \( 1 - 9.55e3T + 2.82e8T^{2} \) |
| 11 | \( 1 - 3.04e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 2.93e4iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 1.45e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 6.37e5iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 3.42e5T + 4.14e13T^{2} \) |
| 29 | \( 1 + 2.17e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 2.54e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 6.66e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 9.83e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 9.92e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 2.47e8T + 5.25e16T^{2} \) |
| 53 | \( 1 - 4.80e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 4.28e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 1.43e9T + 7.13e17T^{2} \) |
| 67 | \( 1 - 1.31e8T + 1.82e18T^{2} \) |
| 71 | \( 1 - 1.02e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 8.16e8iT - 4.29e18T^{2} \) |
| 79 | \( 1 + 1.66e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 4.49e9T + 1.55e19T^{2} \) |
| 89 | \( 1 + 1.63e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 8.94e9iT - 7.37e19T^{2} \) |
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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
−16.08782446278165813631853979240, −14.95728864211855483702455170440, −12.82869585809346425385907012207, −12.02290193992702748890492045928, −11.24270597043032056034598080564, −9.841930295574166678904775235057, −7.08297489846237649394435789975, −5.18866778680284178239879328808, −4.45179996958137590214846136553, −1.36538433527505682475453542726,
0.19293748825419004468164715976, 3.86349978864433143271737998540, 5.48223966586068067609534166862, 6.54125494258251860516430953787, 8.082925884524272629568514740615, 11.07608655631214961375197122699, 11.48820883869584800747903908913, 12.96496205148555048913524596837, 14.63356447445421724609008694937, 15.96482636708935400425629331866