L(s) = 1 | + (−0.366 − 1.69i)3-s + (−1.05 + 1.82i)5-s + (1.43 + 2.49i)7-s + (−2.73 + 1.24i)9-s + (1.21 + 2.10i)11-s + (3.30 − 5.71i)13-s + (3.47 + 1.11i)15-s − 7.56·17-s − 6.25·19-s + (3.69 − 3.35i)21-s + (−2.63 + 4.56i)23-s + (0.275 + 0.476i)25-s + (3.10 + 4.16i)27-s + (−1.57 − 2.73i)29-s + (−1.79 + 3.10i)31-s + ⋯ |
L(s) = 1 | + (−0.211 − 0.977i)3-s + (−0.471 + 0.816i)5-s + (0.543 + 0.942i)7-s + (−0.910 + 0.414i)9-s + (0.365 + 0.633i)11-s + (0.915 − 1.58i)13-s + (0.898 + 0.287i)15-s − 1.83·17-s − 1.43·19-s + (0.805 − 0.731i)21-s + (−0.549 + 0.952i)23-s + (0.0550 + 0.0953i)25-s + (0.597 + 0.801i)27-s + (−0.293 − 0.507i)29-s + (−0.321 + 0.556i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5378422388\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5378422388\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.366 + 1.69i)T \) |
good | 5 | \( 1 + (1.05 - 1.82i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.30 + 5.71i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 + 6.25T + 19T^{2} \) |
| 23 | \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.57 + 2.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.79 - 3.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + (1.74 - 3.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.12 + 5.41i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.32 - 2.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 0.953T + 53T^{2} \) |
| 59 | \( 1 + (4.84 - 8.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.57 - 4.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.949 - 1.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.82T + 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 + (-6.49 - 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.54 + 2.67i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2.95T + 89T^{2} \) |
| 97 | \( 1 + (5.51 + 9.54i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.38592650275856844994187210954, −8.854533160616391439695377644330, −8.455791242183645487471524713436, −7.55155888247441953547279316620, −6.76320345550432589800099591225, −6.03869753808498571892685745268, −5.16867161497701778439081208360, −3.82868322110017800380528466550, −2.63144747319978875988064008885, −1.75812210278473291518889057863,
0.22914289752968810129785239274, 1.90541779685401357790599532853, 3.79646873418236883795536945571, 4.28834951003477563278869060508, 4.78139302397463859966266442329, 6.24773380133106704870688716528, 6.77897088763068903973300933066, 8.355471848281300603861228180607, 8.658720121766754076803404047917, 9.313603308164030175854221811599