| L(s) = 1 | + (4.91 − 3.56i)2-s + (−108. − 334. i)3-s + (−621. + 1.91e3i)4-s + (6.09e3 + 4.42e3i)5-s + (−1.72e3 − 1.25e3i)6-s + (2.00e4 − 6.15e4i)7-s + (7.61e3 + 2.34e4i)8-s + (4.35e4 − 3.16e4i)9-s + 4.56e4·10-s + (5.24e5 − 1.01e5i)11-s + 7.06e5·12-s + (3.95e5 − 2.87e5i)13-s + (−1.21e5 − 3.73e5i)14-s + (8.17e5 − 2.51e6i)15-s + (−3.21e6 − 2.33e6i)16-s + (7.90e6 + 5.74e6i)17-s + ⋯ |
| L(s) = 1 | + (0.108 − 0.0788i)2-s + (−0.257 − 0.793i)3-s + (−0.303 + 0.933i)4-s + (0.871 + 0.633i)5-s + (−0.0905 − 0.0657i)6-s + (0.449 − 1.38i)7-s + (0.0821 + 0.252i)8-s + (0.245 − 0.178i)9-s + 0.144·10-s + (0.981 − 0.190i)11-s + 0.819·12-s + (0.295 − 0.214i)13-s + (−0.0603 − 0.185i)14-s + (0.277 − 0.855i)15-s + (−0.765 − 0.556i)16-s + (1.34 + 0.980i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(1.91413 - 0.449071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.91413 - 0.449071i\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + (-5.24e5 + 1.01e5i)T \) |
| good | 2 | \( 1 + (-4.91 + 3.56i)T + (632. - 1.94e3i)T^{2} \) |
| 3 | \( 1 + (108. + 334. i)T + (-1.43e5 + 1.04e5i)T^{2} \) |
| 5 | \( 1 + (-6.09e3 - 4.42e3i)T + (1.50e7 + 4.64e7i)T^{2} \) |
| 7 | \( 1 + (-2.00e4 + 6.15e4i)T + (-1.59e9 - 1.16e9i)T^{2} \) |
| 13 | \( 1 + (-3.95e5 + 2.87e5i)T + (5.53e11 - 1.70e12i)T^{2} \) |
| 17 | \( 1 + (-7.90e6 - 5.74e6i)T + (1.05e13 + 3.25e13i)T^{2} \) |
| 19 | \( 1 + (-8.92e5 - 2.74e6i)T + (-9.42e13 + 6.84e13i)T^{2} \) |
| 23 | \( 1 + 1.93e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + (3.52e7 - 1.08e8i)T + (-9.87e15 - 7.17e15i)T^{2} \) |
| 31 | \( 1 + (-2.40e8 + 1.74e8i)T + (7.85e15 - 2.41e16i)T^{2} \) |
| 37 | \( 1 + (1.59e8 - 4.92e8i)T + (-1.43e17 - 1.04e17i)T^{2} \) |
| 41 | \( 1 + (1.81e8 + 5.60e8i)T + (-4.45e17 + 3.23e17i)T^{2} \) |
| 43 | \( 1 + 8.27e8T + 9.29e17T^{2} \) |
| 47 | \( 1 + (-9.32e7 - 2.86e8i)T + (-2.00e18 + 1.45e18i)T^{2} \) |
| 53 | \( 1 + (3.74e8 - 2.72e8i)T + (2.86e18 - 8.81e18i)T^{2} \) |
| 59 | \( 1 + (-9.01e8 + 2.77e9i)T + (-2.43e19 - 1.77e19i)T^{2} \) |
| 61 | \( 1 + (3.31e9 + 2.40e9i)T + (1.34e19 + 4.13e19i)T^{2} \) |
| 67 | \( 1 + 1.00e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + (2.16e9 + 1.57e9i)T + (7.14e19 + 2.19e20i)T^{2} \) |
| 73 | \( 1 + (9.23e9 - 2.84e10i)T + (-2.53e20 - 1.84e20i)T^{2} \) |
| 79 | \( 1 + (4.83e7 - 3.51e7i)T + (2.31e20 - 7.11e20i)T^{2} \) |
| 83 | \( 1 + (-5.61e9 - 4.07e9i)T + (3.97e20 + 1.22e21i)T^{2} \) |
| 89 | \( 1 - 7.09e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + (2.75e10 - 2.00e10i)T + (2.21e21 - 6.80e21i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.51915647445451438990608024493, −16.92436683153812592821470750873, −14.26387717060602315448545175105, −13.36966385078602613812904754168, −11.94454372807924414859415100951, −10.16332789362994262975753358181, −7.86114490908009758189516334829, −6.49809628477271713649473497182, −3.77426772696748136693491123225, −1.32081276811230035072134168673,
1.55902731911529118188677422907, 4.82654108347817407093006941167, 5.82486897027151399631223817945, 9.084387869883317488081870776820, 9.946597797504419051541399384719, 11.85729621970517193404381080006, 13.79642357800595843392157075123, 15.07255373407448745242584674736, 16.30786999509145001062483953308, 17.88273663690641830157769925802
