| L(s) = 1 | + (−2.42 − 0.127i)2-s + (−1.28 − 1.58i)3-s + (3.88 + 0.408i)4-s + (−2.16 + 0.556i)5-s + (2.92 + 4.02i)6-s + (−2.31 − 1.28i)7-s + (−4.57 − 0.724i)8-s + (−0.245 + 1.15i)9-s + (5.32 − 1.07i)10-s + (−2.02 + 0.429i)11-s + (−4.34 − 6.69i)12-s + (4.81 + 2.45i)13-s + (5.44 + 3.41i)14-s + (3.67 + 2.72i)15-s + (3.36 + 0.715i)16-s + (1.96 + 5.11i)17-s + ⋯ |
| L(s) = 1 | + (−1.71 − 0.0899i)2-s + (−0.743 − 0.917i)3-s + (1.94 + 0.204i)4-s + (−0.968 + 0.249i)5-s + (1.19 + 1.64i)6-s + (−0.873 − 0.486i)7-s + (−1.61 − 0.256i)8-s + (−0.0819 + 0.385i)9-s + (1.68 − 0.340i)10-s + (−0.609 + 0.129i)11-s + (−1.25 − 1.93i)12-s + (1.33 + 0.679i)13-s + (1.45 + 0.912i)14-s + (0.948 + 0.703i)15-s + (0.841 + 0.178i)16-s + (0.475 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.136465 + 0.0901561i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.136465 + 0.0901561i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.16 - 0.556i)T \) |
| 7 | \( 1 + (2.31 + 1.28i)T \) |
| good | 2 | \( 1 + (2.42 + 0.127i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (1.28 + 1.58i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (2.02 - 0.429i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-4.81 - 2.45i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.96 - 5.11i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.242 - 2.30i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.146 - 2.80i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-3.75 + 5.16i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.88 - 6.47i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.340 + 0.221i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (8.72 - 2.83i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.92 + 3.92i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.50 + 0.960i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (3.15 - 2.55i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.0263 + 0.0292i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (5.29 - 4.76i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (8.24 - 3.16i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (-9.94 - 7.22i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.53 - 2.35i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-3.25 - 7.31i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.44 + 9.13i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-1.60 - 1.78i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-2.60 - 16.4i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.46135440305604081776894913046, −11.67178274629527384848632400884, −10.79869938675189705596957221412, −10.05551945444415147728907374156, −8.653088168437889407767253997128, −7.80855959971666186834574369762, −6.89692878562544145329192531952, −6.19436644400424944744208880592, −3.57727018903007287596511433288, −1.33516124959697652853133305730,
0.31110801661644452734045267708, 3.21274095076408174112203885970, 5.06630297806791959626788752488, 6.41275588169344664740258030817, 7.69061251192177752349791060576, 8.632183779376653668804668780328, 9.525912894398844344211134888102, 10.47978074343920972817977917677, 11.11553608539102825181674122747, 11.95754362571419543701771901149
