| L(s) = 1 | + (0.0218 + 0.416i)2-s + (−0.968 − 0.784i)3-s + (1.81 − 0.190i)4-s + (−1.17 + 1.90i)5-s + (0.305 − 0.420i)6-s + (2.40 + 1.10i)7-s + (0.249 + 1.57i)8-s + (−0.300 − 1.41i)9-s + (−0.817 − 0.447i)10-s + (4.07 + 0.866i)11-s + (−1.90 − 1.23i)12-s + (−0.207 − 0.406i)13-s + (−0.409 + 1.02i)14-s + (2.63 − 0.922i)15-s + (2.92 − 0.621i)16-s + (1.05 + 0.406i)17-s + ⋯ |
| L(s) = 1 | + (0.0154 + 0.294i)2-s + (−0.559 − 0.452i)3-s + (0.908 − 0.0954i)4-s + (−0.525 + 0.851i)5-s + (0.124 − 0.171i)6-s + (0.907 + 0.419i)7-s + (0.0881 + 0.556i)8-s + (−0.100 − 0.471i)9-s + (−0.258 − 0.141i)10-s + (1.22 + 0.261i)11-s + (−0.551 − 0.357i)12-s + (−0.0574 − 0.112i)13-s + (−0.109 + 0.273i)14-s + (0.679 − 0.238i)15-s + (0.730 − 0.155i)16-s + (0.256 + 0.0984i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18662 + 0.259981i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18662 + 0.259981i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.17 - 1.90i)T \) |
| 7 | \( 1 + (-2.40 - 1.10i)T \) |
| good | 2 | \( 1 + (-0.0218 - 0.416i)T + (-1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (0.968 + 0.784i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.07 - 0.866i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.207 + 0.406i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-1.05 - 0.406i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.0282 - 0.269i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (8.97 - 0.470i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.26 - 1.73i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.76 + 6.20i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-0.997 + 1.53i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (6.60 + 2.14i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (6.90 + 6.90i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.00617 - 0.0160i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (-4.26 + 5.26i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (7.19 + 7.98i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (-8.00 - 7.20i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (0.673 - 1.75i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (7.24 - 5.26i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.36 + 2.83i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (6.01 - 13.5i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-13.2 + 2.10i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-2.73 + 3.03i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (8.19 + 1.29i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.20117826636658972309355664522, −11.80807699940979154322188212368, −11.19179729203601238864919503280, −10.00308249630118680944061527202, −8.386885421880298234792861946057, −7.35497846202180216750237254542, −6.52911073940013708220371541149, −5.66871551522385707293700420333, −3.76783614175744095796339722594, −1.92906118178692000715051858750,
1.58442468052377468873378291920, 3.82562560963197697514190343370, 4.86581049079198860583194222877, 6.19439153835276441694509589968, 7.58278978269405264890709783935, 8.455573850504712149255180311817, 9.932837540424910646841203360163, 10.89531506738329251525403806539, 11.78574056503508848468179905181, 12.03455423870353060796407730186
