| L(s) = 1 | + (−0.638 + 2.63i)2-s + (−0.327 + 0.945i)3-s + (−4.73 − 2.44i)4-s + (−0.0914 − 0.0366i)5-s + (−2.27 − 1.46i)6-s + (−1.45 − 2.20i)7-s + (5.90 − 6.81i)8-s + (−0.786 − 0.618i)9-s + (0.154 − 0.217i)10-s + (−1.01 − 4.17i)11-s + (3.85 − 3.67i)12-s + (2.68 + 5.87i)13-s + (6.73 − 2.43i)14-s + (0.0645 − 0.0744i)15-s + (7.97 + 11.2i)16-s + (0.164 − 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.451 + 1.86i)2-s + (−0.188 + 0.545i)3-s + (−2.36 − 1.22i)4-s + (−0.0409 − 0.0163i)5-s + (−0.929 − 0.597i)6-s + (−0.551 − 0.834i)7-s + (2.08 − 2.40i)8-s + (−0.262 − 0.206i)9-s + (0.0489 − 0.0687i)10-s + (−0.305 − 1.25i)11-s + (1.11 − 1.06i)12-s + (0.744 + 1.62i)13-s + (1.80 − 0.649i)14-s + (0.0166 − 0.0192i)15-s + (1.99 + 2.80i)16-s + (0.0399 − 0.838i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.784 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.557258 + 0.193459i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.557258 + 0.193459i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.327 - 0.945i)T \) |
| 7 | \( 1 + (1.45 + 2.20i)T \) |
| 23 | \( 1 + (0.209 + 4.79i)T \) |
| good | 2 | \( 1 + (0.638 - 2.63i)T + (-1.77 - 0.916i)T^{2} \) |
| 5 | \( 1 + (0.0914 + 0.0366i)T + (3.61 + 3.45i)T^{2} \) |
| 11 | \( 1 + (1.01 + 4.17i)T + (-9.77 + 5.04i)T^{2} \) |
| 13 | \( 1 + (-2.68 - 5.87i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.164 + 3.45i)T + (-16.9 - 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.00286 - 0.0600i)T + (-18.9 + 1.80i)T^{2} \) |
| 29 | \( 1 + (-4.91 - 3.15i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.01 - 0.388i)T + (28.7 + 11.5i)T^{2} \) |
| 37 | \( 1 + (0.876 + 0.689i)T + (8.72 + 35.9i)T^{2} \) |
| 41 | \( 1 + (1.14 + 7.98i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (7.14 + 8.25i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (-5.45 + 9.44i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.72 + 0.260i)T + (52.0 - 10.0i)T^{2} \) |
| 59 | \( 1 + (3.58 - 5.03i)T + (-19.2 - 55.7i)T^{2} \) |
| 61 | \( 1 + (0.569 + 1.64i)T + (-47.9 + 37.7i)T^{2} \) |
| 67 | \( 1 + (-7.82 - 7.45i)T + (3.18 + 66.9i)T^{2} \) |
| 71 | \( 1 + (4.36 - 1.28i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (2.94 + 1.51i)T + (42.3 + 59.4i)T^{2} \) |
| 79 | \( 1 + (2.97 + 0.284i)T + (77.5 + 14.9i)T^{2} \) |
| 83 | \( 1 + (-1.54 + 10.7i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.297 + 0.0573i)T + (82.6 - 33.0i)T^{2} \) |
| 97 | \( 1 + (1.53 + 10.6i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.60044628884469338726339881742, −9.993214183456929362600796000129, −8.830102308402820740984770812798, −8.570500278148995540467841521021, −7.18317616456121613032101749988, −6.57428629679210376824362788008, −5.74795386507759372549200750605, −4.61274758331489210543413825597, −3.74780951171443784815230854385, −0.47402238926178426649130055929,
1.38965061592127132541390435890, 2.59878536767688988124906174612, 3.50539930617258034972135023298, 4.91567924392931052946105548366, 6.04801910768782231462261565547, 7.79008937045168725851604201316, 8.318481265379397952163121005672, 9.523576032010842572147241151677, 10.04198633809247119276154895240, 10.94078013708708376314136635139
