| L(s) = 1 | + (0.000898 − 0.0171i)2-s + (0.879 − 0.711i)3-s + (1.98 + 0.209i)4-s + (1.43 − 1.71i)5-s + (−0.0114 − 0.0157i)6-s + (−2.51 − 0.822i)7-s + (0.0107 − 0.0678i)8-s + (−0.357 + 1.68i)9-s + (−0.0281 − 0.0261i)10-s + (−4.96 + 1.05i)11-s + (1.89 − 1.23i)12-s + (0.145 − 0.286i)13-s + (−0.0163 + 0.0423i)14-s + (0.0372 − 2.52i)15-s + (3.91 + 0.831i)16-s + (3.48 − 1.33i)17-s + ⋯ |
| L(s) = 1 | + (0.000635 − 0.0121i)2-s + (0.507 − 0.410i)3-s + (0.994 + 0.104i)4-s + (0.640 − 0.767i)5-s + (−0.00466 − 0.00641i)6-s + (−0.950 − 0.310i)7-s + (0.00379 − 0.0239i)8-s + (−0.119 + 0.561i)9-s + (−0.00890 − 0.00825i)10-s + (−1.49 + 0.317i)11-s + (0.547 − 0.355i)12-s + (0.0404 − 0.0793i)13-s + (−0.00437 + 0.0113i)14-s + (0.00961 − 0.652i)15-s + (0.977 + 0.207i)16-s + (0.846 − 0.324i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49199 - 0.408863i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49199 - 0.408863i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.43 + 1.71i)T \) |
| 7 | \( 1 + (2.51 + 0.822i)T \) |
| good | 2 | \( 1 + (-0.000898 + 0.0171i)T + (-1.98 - 0.209i)T^{2} \) |
| 3 | \( 1 + (-0.879 + 0.711i)T + (0.623 - 2.93i)T^{2} \) |
| 11 | \( 1 + (4.96 - 1.05i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.145 + 0.286i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.48 + 1.33i)T + (12.6 - 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 6.64i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (3.41 + 0.178i)T + (22.8 + 2.40i)T^{2} \) |
| 29 | \( 1 + (-1.99 + 2.74i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.305 + 0.686i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.79 - 2.76i)T + (-15.0 + 33.8i)T^{2} \) |
| 41 | \( 1 + (7.88 - 2.56i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.99 - 3.99i)T - 43iT^{2} \) |
| 47 | \( 1 + (-2.25 + 5.87i)T + (-34.9 - 31.4i)T^{2} \) |
| 53 | \( 1 + (7.10 + 8.77i)T + (-11.0 + 51.8i)T^{2} \) |
| 59 | \( 1 + (-6.39 + 7.10i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.03 - 1.83i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (2.33 + 6.09i)T + (-49.7 + 44.8i)T^{2} \) |
| 71 | \( 1 + (3.66 + 2.66i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.47 + 5.50i)T + (29.6 + 66.6i)T^{2} \) |
| 79 | \( 1 + (-3.33 - 7.49i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-1.28 - 0.203i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (0.663 + 0.736i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-12.2 + 1.94i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.78721594799821032663945749194, −11.89911602361444792361146723901, −10.26504654039952296917924581341, −9.989942233781118261312388872584, −8.202982629885491898957629420742, −7.67046343440579764928264903865, −6.29735426625348077707173435354, −5.25020036691425434098906773826, −3.16147134406834706492569209936, −1.93646819375693468704640729691,
2.61475271533266398075203215745, 3.23969300787659928651361902027, 5.56493415795121676656620581182, 6.44671199583088401110821930309, 7.47830788343433934619275486625, 8.907294465428069008298665757443, 10.04934079474519669448816629675, 10.51934637594232292451721761370, 11.74086118890894284955963938436, 12.84426385777458488668130478554
