| L(s) = 1 | + (−0.261 + 0.513i)2-s + (0.120 + 0.760i)3-s + (0.980 + 1.34i)4-s + (−1.76 − 1.36i)5-s + (−0.421 − 0.136i)6-s + (1.17 + 0.186i)7-s + (−2.08 + 0.330i)8-s + (2.28 − 0.743i)9-s + (1.16 − 0.550i)10-s + (−0.502 − 3.27i)11-s + (−0.908 + 0.908i)12-s + (−5.41 − 2.75i)13-s + (−0.403 + 0.555i)14-s + (0.825 − 1.51i)15-s + (−0.655 + 2.01i)16-s + (0.330 − 0.168i)17-s + ⋯ |
| L(s) = 1 | + (−0.184 + 0.362i)2-s + (0.0695 + 0.438i)3-s + (0.490 + 0.674i)4-s + (−0.791 − 0.611i)5-s + (−0.172 − 0.0559i)6-s + (0.445 + 0.0705i)7-s + (−0.737 + 0.116i)8-s + (0.763 − 0.247i)9-s + (0.368 − 0.174i)10-s + (−0.151 − 0.988i)11-s + (−0.262 + 0.262i)12-s + (−1.50 − 0.765i)13-s + (−0.107 + 0.148i)14-s + (0.213 − 0.389i)15-s + (−0.163 + 0.504i)16-s + (0.0801 − 0.0408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.738277 + 0.333224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738277 + 0.333224i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.76 + 1.36i)T \) |
| 11 | \( 1 + (0.502 + 3.27i)T \) |
| good | 2 | \( 1 + (0.261 - 0.513i)T + (-1.17 - 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.120 - 0.760i)T + (-2.85 + 0.927i)T^{2} \) |
| 7 | \( 1 + (-1.17 - 0.186i)T + (6.65 + 2.16i)T^{2} \) |
| 13 | \( 1 + (5.41 + 2.75i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.330 + 0.168i)T + (9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.11 - 0.809i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-3.48 - 3.48i)T + 23iT^{2} \) |
| 29 | \( 1 + (4.95 - 3.60i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.764 + 2.35i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.465 - 2.93i)T + (-35.1 - 11.4i)T^{2} \) |
| 41 | \( 1 + (3.60 - 4.95i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-6.75 + 6.75i)T - 43iT^{2} \) |
| 47 | \( 1 + (1.26 - 0.199i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (-0.185 + 0.363i)T + (-31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-0.110 - 0.151i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (0.649 + 0.211i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-7.14 + 7.14i)T - 67iT^{2} \) |
| 71 | \( 1 + (-0.319 + 0.983i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.715 + 4.51i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (-3.59 - 11.0i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-5.16 - 10.1i)T + (-48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 8.04iT - 89T^{2} \) |
| 97 | \( 1 + (-7.52 - 3.83i)T + (57.0 + 78.4i)T^{2} \) |
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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
−15.55196528483753144229930700224, −14.88101611477977938100491300300, −12.99352743719574080358343410173, −12.09851512397631091060124262723, −11.02380155404627652560807891873, −9.381735464886320909418381026164, −8.107058564439821617868113746097, −7.25032399552082226870441340822, −5.15726334648729579152720252335, −3.45552171887150784266093300837,
2.23701452163949610681131906161, 4.68686299969765933047277472462, 6.87470734013728367665563832871, 7.53483875237760741378546355654, 9.575765427438229473013973658590, 10.61283937000209686574001720657, 11.72375437909449330904455988577, 12.60226283157882052521678759037, 14.40296267572224139518533038774, 15.00597945975132628554837847930
