| L(s) = 1 | + (0.622 − 1.26i)2-s + (−1.30 + 0.695i)3-s + (−1.22 − 1.58i)4-s + (−2.50 − 3.05i)5-s + (0.0725 + 2.08i)6-s + (1.90 − 1.27i)7-s + (−2.77 + 0.568i)8-s + (−0.456 + 0.683i)9-s + (−5.43 + 1.27i)10-s + (4.47 − 1.35i)11-s + (2.69 + 1.20i)12-s + (−2.41 − 1.98i)13-s + (−0.428 − 3.20i)14-s + (5.38 + 2.22i)15-s + (−1.00 + 3.87i)16-s + (4.12 − 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.440 − 0.897i)2-s + (−0.751 + 0.401i)3-s + (−0.611 − 0.790i)4-s + (−1.11 − 1.36i)5-s + (0.0296 + 0.851i)6-s + (0.718 − 0.480i)7-s + (−0.979 + 0.201i)8-s + (−0.152 + 0.227i)9-s + (−1.71 + 0.404i)10-s + (1.34 − 0.408i)11-s + (0.777 + 0.348i)12-s + (−0.671 − 0.550i)13-s + (−0.114 − 0.856i)14-s + (1.38 + 0.575i)15-s + (−0.250 + 0.968i)16-s + (0.999 − 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.321062 - 0.777414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.321062 - 0.777414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.622 + 1.26i)T \) |
| good | 3 | \( 1 + (1.30 - 0.695i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (2.50 + 3.05i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-1.90 + 1.27i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (-4.47 + 1.35i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.41 + 1.98i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.12 + 1.70i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.371 - 3.77i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-1.98 + 0.395i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.870 + 2.86i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-0.377 - 0.377i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.87 - 0.184i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (2.35 + 11.8i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.04 - 1.09i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (-0.807 - 1.95i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (2.23 + 7.35i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (9.34 - 7.66i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.03 - 7.55i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (-3.52 - 6.59i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (-5.81 - 8.70i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (4.18 + 2.79i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.189 + 0.457i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-8.89 + 0.875i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.75 - 1.14i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-0.489 - 0.489i)T + 97iT^{2} \) |
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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
−12.48035645129152423454898630837, −11.88852944538788166458960947488, −11.28025587543673172061008802548, −10.12841646376103925564081430455, −8.874094101272039874408803569337, −7.80244891864531065722926492264, −5.60849427243303301977558319511, −4.73490776526187383331447956758, −3.82894792971576901339780236411, −0.940303809647705085763462022330,
3.35730293431655633083154830900, 4.76452504847315159056462544229, 6.33905544354038744405886643353, 6.96749485721432792935824559411, 7.937564304256754245531405443021, 9.331181029364827452402128355222, 11.18047805514082583814564248867, 11.81644367547469096838689734256, 12.39945845688572573964554368453, 14.22842387332282972527112178328
