| L(s) = 1 | + (1.38 − 0.278i)2-s + (−2.14 − 2.28i)3-s + (1.84 − 0.771i)4-s + (−1.12 − 2.49i)5-s + (−3.60 − 2.57i)6-s + (2.40 − 1.10i)7-s + (2.34 − 1.58i)8-s + (−0.444 + 6.78i)9-s + (−2.25 − 3.14i)10-s + (−1.32 − 2.13i)11-s + (−5.71 − 2.56i)12-s + (−0.576 − 5.85i)13-s + (3.02 − 2.19i)14-s + (−3.27 + 7.91i)15-s + (2.81 − 2.84i)16-s + (5.78 + 0.762i)17-s + ⋯ |
| L(s) = 1 | + (0.980 − 0.196i)2-s + (−1.23 − 1.31i)3-s + (0.922 − 0.385i)4-s + (−0.504 − 1.11i)5-s + (−1.47 − 1.05i)6-s + (0.908 − 0.417i)7-s + (0.828 − 0.559i)8-s + (−0.148 + 2.26i)9-s + (−0.714 − 0.993i)10-s + (−0.399 − 0.642i)11-s + (−1.64 − 0.740i)12-s + (−0.159 − 1.62i)13-s + (0.809 − 0.587i)14-s + (−0.846 + 2.04i)15-s + (0.702 − 0.711i)16-s + (1.40 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0264424 - 1.86886i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0264424 - 1.86886i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.38 + 0.278i)T \) |
| 7 | \( 1 + (-2.40 + 1.10i)T \) |
| good | 3 | \( 1 + (2.14 + 2.28i)T + (-0.196 + 2.99i)T^{2} \) |
| 5 | \( 1 + (1.12 + 2.49i)T + (-3.29 + 3.75i)T^{2} \) |
| 11 | \( 1 + (1.32 + 2.13i)T + (-4.86 + 9.86i)T^{2} \) |
| 13 | \( 1 + (0.576 + 5.85i)T + (-12.7 + 2.53i)T^{2} \) |
| 17 | \( 1 + (-5.78 - 0.762i)T + (16.4 + 4.39i)T^{2} \) |
| 19 | \( 1 + (0.615 - 3.72i)T + (-17.9 - 6.10i)T^{2} \) |
| 23 | \( 1 + (0.178 + 0.203i)T + (-3.00 + 22.8i)T^{2} \) |
| 29 | \( 1 + (-2.96 - 5.54i)T + (-16.1 + 24.1i)T^{2} \) |
| 31 | \( 1 + (9.11 - 2.44i)T + (26.8 - 15.5i)T^{2} \) |
| 37 | \( 1 + (3.07 - 8.17i)T + (-27.8 - 24.3i)T^{2} \) |
| 41 | \( 1 + (-1.44 - 7.28i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-3.93 + 1.19i)T + (35.7 - 23.8i)T^{2} \) |
| 47 | \( 1 + (-2.38 + 3.11i)T + (-12.1 - 45.3i)T^{2} \) |
| 53 | \( 1 + (-1.61 - 2.59i)T + (-23.4 + 47.5i)T^{2} \) |
| 59 | \( 1 + (2.07 - 2.89i)T + (-18.9 - 55.8i)T^{2} \) |
| 61 | \( 1 + (-6.15 + 1.43i)T + (54.7 - 26.9i)T^{2} \) |
| 67 | \( 1 + (3.62 + 3.87i)T + (-4.38 + 66.8i)T^{2} \) |
| 71 | \( 1 + (1.74 + 2.61i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.11 + 1.04i)T + (44.4 - 57.9i)T^{2} \) |
| 79 | \( 1 + (0.450 + 3.41i)T + (-76.3 + 20.4i)T^{2} \) |
| 83 | \( 1 + (0.614 - 0.748i)T + (-16.1 - 81.4i)T^{2} \) |
| 89 | \( 1 + (0.0464 + 0.136i)T + (-70.6 + 54.1i)T^{2} \) |
| 97 | \( 1 + (-5.09 + 5.09i)T - 97iT^{2} \) |
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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
−10.36754182336981971931114986154, −8.236375013017651329161263574133, −7.889589549569278380432203425465, −7.10196161193806585439547294168, −5.76749481883480188190785953428, −5.43808066604264243491841878277, −4.72210878027190527544524162951, −3.27582276628516073215806171314, −1.52329975694629492461673653470, −0.817494661477137853279736529429,
2.32882595504825647650643471072, 3.73057771144334149334165090757, 4.35302016916722006964845386706, 5.20842447294672413359088023959, 5.88556534299915509248784271215, 6.96813401233804796639716257376, 7.52822356674756815693441471078, 9.066845457239195666737459417168, 10.04409538643149445169575478551, 10.96447958637342515028622997365
