| L(s) = 1 | + (−0.695 − 2.00i)2-s + (−0.888 − 0.458i)3-s + (−1.98 + 1.55i)4-s + (1.53 + 0.146i)5-s + (−0.302 + 2.10i)6-s + (0.278 − 2.63i)7-s + (0.930 + 0.597i)8-s + (0.580 + 0.814i)9-s + (−0.773 − 3.18i)10-s + (1.64 − 4.76i)11-s + (2.47 − 0.476i)12-s + (5.59 + 1.64i)13-s + (−5.48 + 1.26i)14-s + (−1.29 − 0.834i)15-s + (−0.633 + 2.61i)16-s + (1.71 − 0.687i)17-s + ⋯ |
| L(s) = 1 | + (−0.491 − 1.42i)2-s + (−0.513 − 0.264i)3-s + (−0.990 + 0.778i)4-s + (0.687 + 0.0656i)5-s + (−0.123 + 0.859i)6-s + (0.105 − 0.994i)7-s + (0.328 + 0.211i)8-s + (0.193 + 0.271i)9-s + (−0.244 − 1.00i)10-s + (0.497 − 1.43i)11-s + (0.714 − 0.137i)12-s + (1.55 + 0.455i)13-s + (−1.46 + 0.339i)14-s + (−0.335 − 0.215i)15-s + (−0.158 + 0.653i)16-s + (0.416 − 0.166i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0834143 + 0.943315i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0834143 + 0.943315i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.888 + 0.458i)T \) |
| 7 | \( 1 + (-0.278 + 2.63i)T \) |
| 23 | \( 1 + (4.08 + 2.50i)T \) |
| good | 2 | \( 1 + (0.695 + 2.00i)T + (-1.57 + 1.23i)T^{2} \) |
| 5 | \( 1 + (-1.53 - 0.146i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (-1.64 + 4.76i)T + (-8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (-5.59 - 1.64i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-1.71 + 0.687i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (5.97 + 2.39i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 2.14i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (0.253 - 5.31i)T + (-30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-2.49 - 3.50i)T + (-12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (0.514 + 1.12i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (4.61 - 2.96i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.76 - 4.54i)T + (2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-0.0726 - 0.299i)T + (-52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (1.00 - 0.517i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (3.30 + 0.636i)T + (62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-9.95 + 11.4i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-9.61 + 7.55i)T + (17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (9.47 - 9.03i)T + (3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (2.42 - 5.31i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (0.691 + 14.5i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (2.45 + 5.38i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.69550720073424136045322398789, −9.998718406522337301375317156559, −8.880052871118040978756353253847, −8.227665205389199315981660997431, −6.52563792336729882751475906643, −6.06314654473609323892442778515, −4.28616784967366452359348866239, −3.37622621819460366707020870143, −1.82338999996866400473994275680, −0.75597651656929718853444844346,
1.91697706848832931617461893266, 4.05359004028564999378716294116, 5.41725885902664025046924087374, 5.96778102836653040045570034829, 6.62324625188494657621385732236, 7.87721499332261876210844955239, 8.657945819364725091297700343446, 9.513337152170320745466172484774, 10.16050631913917803170803310000, 11.41782412059426521220981751103
