| L(s) = 1 | + (0.225 + 1.39i)2-s + (1.39 + 2.77i)3-s + (−1.89 + 0.629i)4-s + (−3.12 + 1.38i)5-s + (−3.56 + 2.58i)6-s + (2.65 + 1.58i)7-s + (−1.30 − 2.50i)8-s + (−3.98 + 5.36i)9-s + (−2.64 − 4.05i)10-s + (2.07 − 2.65i)11-s + (−4.40 − 4.39i)12-s + (2.27 + 2.38i)13-s + (−1.62 + 4.06i)14-s + (−8.22 − 6.75i)15-s + (3.20 − 2.39i)16-s + (−0.900 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (0.159 + 0.987i)2-s + (0.807 + 1.60i)3-s + (−0.949 + 0.314i)4-s + (−1.39 + 0.619i)5-s + (−1.45 + 1.05i)6-s + (1.00 + 0.600i)7-s + (−0.462 − 0.886i)8-s + (−1.32 + 1.78i)9-s + (−0.835 − 1.28i)10-s + (0.624 − 0.799i)11-s + (−1.27 − 1.26i)12-s + (0.630 + 0.662i)13-s + (−0.433 + 1.08i)14-s + (−2.12 − 1.74i)15-s + (0.801 − 0.597i)16-s + (−0.218 − 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.486969 - 1.45959i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.486969 - 1.45959i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.225 - 1.39i)T \) |
| good | 3 | \( 1 + (-1.39 - 2.77i)T + (-1.78 + 2.40i)T^{2} \) |
| 5 | \( 1 + (3.12 - 1.38i)T + (3.35 - 3.70i)T^{2} \) |
| 7 | \( 1 + (-2.65 - 1.58i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-2.07 + 2.65i)T + (-2.67 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.27 - 2.38i)T + (-0.637 + 12.9i)T^{2} \) |
| 17 | \( 1 + (0.900 + 1.09i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (-7.18 + 1.24i)T + (17.8 - 6.40i)T^{2} \) |
| 23 | \( 1 + (1.89 + 0.894i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (-1.88 + 3.32i)T + (-14.9 - 24.8i)T^{2} \) |
| 31 | \( 1 + (6.73 - 1.33i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (5.84 - 9.22i)T + (-15.8 - 33.4i)T^{2} \) |
| 41 | \( 1 + (5.36 + 5.92i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-3.71 + 1.22i)T + (34.5 - 25.6i)T^{2} \) |
| 47 | \( 1 + (-11.5 - 6.16i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (0.886 + 1.56i)T + (-27.2 + 45.4i)T^{2} \) |
| 59 | \( 1 + (-0.833 - 0.793i)T + (2.89 + 58.9i)T^{2} \) |
| 61 | \( 1 + (-4.54 + 0.335i)T + (60.3 - 8.95i)T^{2} \) |
| 67 | \( 1 + (-4.43 + 5.13i)T + (-9.83 - 66.2i)T^{2} \) |
| 71 | \( 1 + (10.6 + 1.57i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (10.5 - 6.34i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (0.339 + 0.103i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-0.901 - 1.42i)T + (-35.4 + 75.0i)T^{2} \) |
| 89 | \( 1 + (-10.1 + 4.79i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (-4.70 - 3.14i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.47025786257194478732061193069, −10.47312588541900875652098273008, −9.233099136127467084442387525491, −8.693210962469873524700771734885, −8.058692507923642207625564994455, −7.12144940760140925884374511085, −5.62215097475471083022110933493, −4.65361065873642170432565912158, −3.84630084145179551004599913448, −3.16798797577469307227044846703,
0.901530505507249839564206880420, 1.73983091368830593493922041592, 3.37508492480222954999227691242, 4.13863672779118015469762506381, 5.47861384647870919683720854536, 7.23931975323169661200476465506, 7.72251204697782892801370961778, 8.481719100019074033825091834246, 9.179821503605972804328344753257, 10.68058961258094341622201760055
