| L(s) = 1 | + (−0.643 + 1.25i)2-s + (0.195 + 1.72i)3-s + (−1.17 − 1.62i)4-s + (−0.672 + 1.98i)5-s + (−2.29 − 0.860i)6-s + (−1.53 + 1.99i)7-s + (2.79 − 0.433i)8-s + (−2.92 + 0.673i)9-s + (−2.06 − 2.12i)10-s + (2.43 + 0.159i)11-s + (2.55 − 2.33i)12-s + (−2.28 + 2.61i)13-s + (−1.52 − 3.21i)14-s + (−3.54 − 0.770i)15-s + (−1.25 + 3.79i)16-s + (−1.22 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.890i)2-s + (0.112 + 0.993i)3-s + (−0.586 − 0.810i)4-s + (−0.300 + 0.886i)5-s + (−0.936 − 0.351i)6-s + (−0.578 + 0.753i)7-s + (0.988 − 0.153i)8-s + (−0.974 + 0.224i)9-s + (−0.652 − 0.671i)10-s + (0.732 + 0.0480i)11-s + (0.738 − 0.673i)12-s + (−0.635 + 0.724i)13-s + (−0.408 − 0.857i)14-s + (−0.914 − 0.198i)15-s + (−0.312 + 0.949i)16-s + (−0.297 − 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.422 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.329147 - 0.516698i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.329147 - 0.516698i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.643 - 1.25i)T \) |
| 3 | \( 1 + (-0.195 - 1.72i)T \) |
| good | 5 | \( 1 + (0.672 - 1.98i)T + (-3.96 - 3.04i)T^{2} \) |
| 7 | \( 1 + (1.53 - 1.99i)T + (-1.81 - 6.76i)T^{2} \) |
| 11 | \( 1 + (-2.43 - 0.159i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (2.28 - 2.61i)T + (-1.69 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.214 + 1.07i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.0185 - 0.0142i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-3.20 + 6.49i)T + (-17.6 - 23.0i)T^{2} \) |
| 31 | \( 1 + (7.56 + 4.36i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 8.22i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.841 + 0.645i)T + (10.6 - 39.6i)T^{2} \) |
| 43 | \( 1 + (0.0881 - 1.34i)T + (-42.6 - 5.61i)T^{2} \) |
| 47 | \( 1 + (0.434 - 1.62i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.30 - 1.95i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-1.32 + 3.91i)T + (-46.8 - 35.9i)T^{2} \) |
| 61 | \( 1 + (-2.46 - 1.21i)T + (37.1 + 48.3i)T^{2} \) |
| 67 | \( 1 + (1.00 + 15.3i)T + (-66.4 + 8.74i)T^{2} \) |
| 71 | \( 1 + (-3.18 - 7.69i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (5.78 - 13.9i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (9.04 + 2.42i)T + (68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (8.05 - 2.73i)T + (65.8 - 50.5i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 4.79i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (1.38 - 0.802i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.16967694813443016257169894144, −10.11010510047006324056472832700, −9.439726461174504987750193037006, −8.920128538708197083759950401265, −7.77920055034392070026367915162, −6.74422730371307681996082131998, −6.05850021058503483034927493660, −4.89576941253617161100952602077, −3.89564944748158266504514668585, −2.55689007117335954615178903043,
0.40668528667694898740492921735, 1.56285423844664659058723404476, 3.07271947769618200650924353844, 4.09290132229737593989161864089, 5.37867107190601914921807214713, 6.87345637265713005210865379922, 7.54464681576035426650870541096, 8.571715361320235626232471394498, 9.079412184097104953756028993297, 10.19493604699120194503642072140
