| L(s) = 1 | + (21.8 − 5.85i)2-s + (−233. + 68.3i)3-s + (443. − 256i)4-s + (2.93e3 − 1.07e3i)5-s + (−4.69e3 + 2.85e3i)6-s + (−2.67e4 + 7.17e3i)7-s + (8.19e3 − 8.19e3i)8-s + (4.97e4 − 3.18e4i)9-s + (5.77e4 − 4.07e4i)10-s + (−3.83e4 + 6.64e4i)11-s + (−8.58e4 + 9.00e4i)12-s + (4.07e5 + 1.09e5i)13-s + (−5.43e5 + 3.13e5i)14-s + (−6.10e5 + 4.52e5i)15-s + (1.31e5 − 2.27e5i)16-s + (−1.71e6 − 1.71e6i)17-s + ⋯ |
| L(s) = 1 | + (0.683 − 0.183i)2-s + (−0.959 + 0.281i)3-s + (0.433 − 0.250i)4-s + (0.938 − 0.345i)5-s + (−0.603 + 0.367i)6-s + (−1.59 + 0.427i)7-s + (0.249 − 0.250i)8-s + (0.841 − 0.539i)9-s + (0.577 − 0.407i)10-s + (−0.238 + 0.412i)11-s + (−0.345 + 0.361i)12-s + (1.09 + 0.294i)13-s + (−1.01 + 0.583i)14-s + (−0.803 + 0.595i)15-s + (0.124 − 0.216i)16-s + (−1.21 − 1.21i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.862 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.13367 - 0.579840i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.13367 - 0.579840i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-21.8 + 5.85i)T \) |
| 3 | \( 1 + (233. - 68.3i)T \) |
| 5 | \( 1 + (-2.93e3 + 1.07e3i)T \) |
| good | 7 | \( 1 + (2.67e4 - 7.17e3i)T + (2.44e8 - 1.41e8i)T^{2} \) |
| 11 | \( 1 + (3.83e4 - 6.64e4i)T + (-1.29e10 - 2.24e10i)T^{2} \) |
| 13 | \( 1 + (-4.07e5 - 1.09e5i)T + (1.19e11 + 6.89e10i)T^{2} \) |
| 17 | \( 1 + (1.71e6 + 1.71e6i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 - 2.67e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 + (-2.68e6 - 7.19e5i)T + (3.58e13 + 2.07e13i)T^{2} \) |
| 29 | \( 1 + (-1.72e7 - 9.95e6i)T + (2.10e14 + 3.64e14i)T^{2} \) |
| 31 | \( 1 + (2.23e7 + 3.86e7i)T + (-4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + (-1.72e6 - 1.72e6i)T + 4.80e15iT^{2} \) |
| 41 | \( 1 + (-9.46e7 - 1.63e8i)T + (-6.71e15 + 1.16e16i)T^{2} \) |
| 43 | \( 1 + (2.74e7 + 1.02e8i)T + (-1.87e16 + 1.08e16i)T^{2} \) |
| 47 | \( 1 + (-3.19e8 + 8.56e7i)T + (4.55e16 - 2.62e16i)T^{2} \) |
| 53 | \( 1 + (-5.00e8 + 5.00e8i)T - 1.74e17iT^{2} \) |
| 59 | \( 1 + (-4.42e8 + 2.55e8i)T + (2.55e17 - 4.42e17i)T^{2} \) |
| 61 | \( 1 + (-7.31e7 + 1.26e8i)T + (-3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-2.95e8 + 1.10e9i)T + (-1.57e18 - 9.11e17i)T^{2} \) |
| 71 | \( 1 - 1.24e9T + 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.63e9 + 2.63e9i)T - 4.29e18iT^{2} \) |
| 79 | \( 1 + (-3.60e9 - 2.07e9i)T + (4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-3.42e8 - 1.27e9i)T + (-1.34e19 + 7.75e18i)T^{2} \) |
| 89 | \( 1 + 4.73e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + (-1.26e10 + 3.39e9i)T + (6.38e19 - 3.68e19i)T^{2} \) |
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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
−12.19071379567837534876374253528, −10.98680192721511860174154650679, −9.891335741539046066338864682691, −9.180218778149919814412953322316, −6.76153663679928086895896416252, −6.12484383793329493951137804219, −5.15258245867409661507897092983, −3.78273348253519253242285385589, −2.27468341062985703786227842144, −0.67052043513244011696651591775,
0.871095856094286373217858467853, 2.56207959968286670191843703265, 3.94813690718713123374300423269, 5.57018461072502093600778014678, 6.38348472610262369242266842283, 6.94552653605731486630097754345, 8.963186617369330883384276334155, 10.50380153722012903538106615132, 10.87950457614699797993535015556, 12.57530089354868622106296530839
