| L(s) = 1 | + (0.990 + 1.00i)2-s + (−2.52 + 1.27i)3-s + (−0.0391 + 1.99i)4-s + (0.365 + 0.825i)5-s + (−3.78 − 1.29i)6-s + (4.31 + 2.58i)7-s + (−2.05 + 1.94i)8-s + (2.96 − 4.00i)9-s + (−0.471 + 1.18i)10-s + (3.05 + 2.38i)11-s + (−2.44 − 5.09i)12-s + (−3.45 + 3.29i)13-s + (1.66 + 6.91i)14-s + (−1.97 − 1.61i)15-s + (−3.99 − 0.156i)16-s + (−1.22 − 1.49i)17-s + ⋯ |
| L(s) = 1 | + (0.700 + 0.714i)2-s + (−1.45 + 0.733i)3-s + (−0.0195 + 0.999i)4-s + (0.163 + 0.369i)5-s + (−1.54 − 0.526i)6-s + (1.63 + 0.977i)7-s + (−0.727 + 0.686i)8-s + (0.989 − 1.33i)9-s + (−0.148 + 0.375i)10-s + (0.919 + 0.717i)11-s + (−0.704 − 1.47i)12-s + (−0.959 + 0.913i)13-s + (0.443 + 1.84i)14-s + (−0.509 − 0.417i)15-s + (−0.999 − 0.0391i)16-s + (−0.297 − 0.362i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0731464 + 1.45475i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0731464 + 1.45475i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.990 - 1.00i)T \) |
| good | 3 | \( 1 + (2.52 - 1.27i)T + (1.78 - 2.40i)T^{2} \) |
| 5 | \( 1 + (-0.365 - 0.825i)T + (-3.35 + 3.70i)T^{2} \) |
| 7 | \( 1 + (-4.31 - 2.58i)T + (3.29 + 6.17i)T^{2} \) |
| 11 | \( 1 + (-3.05 - 2.38i)T + (2.67 + 10.6i)T^{2} \) |
| 13 | \( 1 + (3.45 - 3.29i)T + (0.637 - 12.9i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.49i)T + (-3.31 + 16.6i)T^{2} \) |
| 19 | \( 1 + (1.31 + 7.58i)T + (-17.8 + 6.40i)T^{2} \) |
| 23 | \( 1 + (-5.35 - 2.53i)T + (14.5 + 17.7i)T^{2} \) |
| 29 | \( 1 + (1.50 + 0.853i)T + (14.9 + 24.8i)T^{2} \) |
| 31 | \( 1 + (3.07 - 0.611i)T + (28.6 - 11.8i)T^{2} \) |
| 37 | \( 1 + (4.97 + 3.15i)T + (15.8 + 33.4i)T^{2} \) |
| 41 | \( 1 + (3.07 + 3.38i)T + (-4.01 + 40.8i)T^{2} \) |
| 43 | \( 1 + (-1.47 - 4.47i)T + (-34.5 + 25.6i)T^{2} \) |
| 47 | \( 1 + (-2.18 - 1.16i)T + (26.1 + 39.0i)T^{2} \) |
| 53 | \( 1 + (-2.01 + 1.14i)T + (27.2 - 45.4i)T^{2} \) |
| 59 | \( 1 + (-7.75 + 8.14i)T + (-2.89 - 58.9i)T^{2} \) |
| 61 | \( 1 + (-0.281 - 3.81i)T + (-60.3 + 8.95i)T^{2} \) |
| 67 | \( 1 + (-10.8 - 9.35i)T + (9.83 + 66.2i)T^{2} \) |
| 71 | \( 1 + (2.33 + 0.345i)T + (67.9 + 20.6i)T^{2} \) |
| 73 | \( 1 + (0.00781 - 0.00468i)T + (34.4 - 64.3i)T^{2} \) |
| 79 | \( 1 + (3.52 + 1.06i)T + (65.6 + 43.8i)T^{2} \) |
| 83 | \( 1 + (-0.499 + 0.316i)T + (35.4 - 75.0i)T^{2} \) |
| 89 | \( 1 + (-9.39 + 4.44i)T + (56.4 - 68.7i)T^{2} \) |
| 97 | \( 1 + (7.10 + 4.74i)T + (37.1 + 89.6i)T^{2} \) |
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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
−11.52384736653722345560449670791, −10.89937772551851140703518009095, −9.396418260135103803186953199864, −8.784872951954488198618625952490, −7.16683313575376011854646531140, −6.71035643347748808694581608910, −5.42706700366764166097696559961, −4.88908820085127791290221900980, −4.31859999028771794119547549866, −2.29886915019215747145841638673,
0.906656450306897213857721347332, 1.69823613854950206888292019126, 3.85289246858356175000162803164, 5.00573498020995168069743441462, 5.43382287206760244469087126884, 6.55735784349805719555849891087, 7.48108272259920898374094451920, 8.647513806432978928866783933757, 10.23376622183502398235168712249, 10.79070005471197878371413188918
