| L(s) = 1 | + (0.0194 − 0.118i)2-s + (0.267 − 0.963i)3-s + (1.88 + 0.633i)4-s + (−1.57 − 2.32i)5-s + (−0.109 − 0.0504i)6-s + (1.03 − 0.228i)7-s + (0.224 − 0.423i)8-s + (−0.856 − 0.515i)9-s + (−0.306 + 0.142i)10-s + (−0.225 + 0.171i)11-s + (1.11 − 1.64i)12-s + (0.273 − 0.164i)13-s + (−0.00692 − 0.127i)14-s + (−2.66 + 0.898i)15-s + (3.11 + 2.36i)16-s + (0.728 + 0.160i)17-s + ⋯ |
| L(s) = 1 | + (0.0137 − 0.0839i)2-s + (0.154 − 0.556i)3-s + (0.940 + 0.316i)4-s + (−0.705 − 1.04i)5-s + (−0.0445 − 0.0206i)6-s + (0.392 − 0.0863i)7-s + (0.0793 − 0.149i)8-s + (−0.285 − 0.171i)9-s + (−0.0970 + 0.0449i)10-s + (−0.0680 + 0.0517i)11-s + (0.321 − 0.474i)12-s + (0.0759 − 0.0457i)13-s + (−0.00184 − 0.0341i)14-s + (−0.688 + 0.231i)15-s + (0.778 + 0.592i)16-s + (0.176 + 0.0388i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20467 - 0.610859i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20467 - 0.610859i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.267 + 0.963i)T \) |
| 59 | \( 1 + (-4.51 + 6.21i)T \) |
| good | 2 | \( 1 + (-0.0194 + 0.118i)T + (-1.89 - 0.638i)T^{2} \) |
| 5 | \( 1 + (1.57 + 2.32i)T + (-1.85 + 4.64i)T^{2} \) |
| 7 | \( 1 + (-1.03 + 0.228i)T + (6.35 - 2.93i)T^{2} \) |
| 11 | \( 1 + (0.225 - 0.171i)T + (2.94 - 10.5i)T^{2} \) |
| 13 | \( 1 + (-0.273 + 0.164i)T + (6.08 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-0.728 - 0.160i)T + (15.4 + 7.13i)T^{2} \) |
| 19 | \( 1 + (2.09 - 0.228i)T + (18.5 - 4.08i)T^{2} \) |
| 23 | \( 1 + (-4.27 - 5.03i)T + (-3.72 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.142 - 0.870i)T + (-27.4 + 9.25i)T^{2} \) |
| 31 | \( 1 + (8.02 + 0.872i)T + (30.2 + 6.66i)T^{2} \) |
| 37 | \( 1 + (-2.76 - 5.21i)T + (-20.7 + 30.6i)T^{2} \) |
| 41 | \( 1 + (5.00 - 5.88i)T + (-6.63 - 40.4i)T^{2} \) |
| 43 | \( 1 + (-7.12 - 5.41i)T + (11.5 + 41.4i)T^{2} \) |
| 47 | \( 1 + (4.51 - 6.65i)T + (-17.3 - 43.6i)T^{2} \) |
| 53 | \( 1 + (7.81 + 3.61i)T + (34.3 + 40.3i)T^{2} \) |
| 61 | \( 1 + (0.0621 - 0.379i)T + (-57.8 - 19.4i)T^{2} \) |
| 67 | \( 1 + (-5.29 + 9.98i)T + (-37.5 - 55.4i)T^{2} \) |
| 71 | \( 1 + (-3.19 + 4.71i)T + (-26.2 - 65.9i)T^{2} \) |
| 73 | \( 1 + (0.658 + 12.1i)T + (-72.5 + 7.89i)T^{2} \) |
| 79 | \( 1 + (2.62 + 9.43i)T + (-67.6 + 40.7i)T^{2} \) |
| 83 | \( 1 + (-5.69 - 5.39i)T + (4.49 + 82.8i)T^{2} \) |
| 89 | \( 1 + (0.335 + 2.04i)T + (-84.3 + 28.4i)T^{2} \) |
| 97 | \( 1 + (0.218 - 4.03i)T + (-96.4 - 10.4i)T^{2} \) |
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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
−12.60713057461710340497557590587, −11.60264032404904257591071012061, −10.99014098960588802969236921867, −9.356610144325041572349221078711, −8.137508073041888989192393265450, −7.62243741568081146429905441493, −6.35613093467699599730228148070, −4.87212624481663295358918052588, −3.34372053248028078771039780752, −1.53963738354308464159027612965,
2.47000565749625422618449060306, 3.74895070048317998626919465500, 5.36525402563054752531442907929, 6.71080965802670179262337237021, 7.51216850762547093770137604045, 8.728119170282564657768365332418, 10.20375554975404599233594963068, 10.94861728764431283083636165014, 11.44995778078783809539466762136, 12.66614853936593133095501489503
