| L(s) = 1 | + (−1.49 − 1.42i)2-s + (−0.981 + 0.189i)3-s + (0.107 + 2.25i)4-s + (2.56 − 2.01i)5-s + (1.73 + 1.11i)6-s + (0.200 − 2.63i)7-s + (0.342 − 0.395i)8-s + (0.928 − 0.371i)9-s + (−6.69 − 0.639i)10-s + (2.38 − 2.27i)11-s + (−0.531 − 2.19i)12-s + (2.65 + 5.81i)13-s + (−4.05 + 3.65i)14-s + (−2.13 + 2.46i)15-s + (3.41 − 0.325i)16-s + (6.09 − 3.13i)17-s + ⋯ |
| L(s) = 1 | + (−1.05 − 1.00i)2-s + (−0.566 + 0.109i)3-s + (0.0536 + 1.12i)4-s + (1.14 − 0.901i)5-s + (0.708 + 0.455i)6-s + (0.0757 − 0.997i)7-s + (0.121 − 0.139i)8-s + (0.309 − 0.123i)9-s + (−2.11 − 0.202i)10-s + (0.718 − 0.685i)11-s + (−0.153 − 0.632i)12-s + (0.737 + 1.61i)13-s + (−1.08 + 0.976i)14-s + (−0.551 + 0.636i)15-s + (0.853 − 0.0814i)16-s + (1.47 − 0.761i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.357468 - 0.802791i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.357468 - 0.802791i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.981 - 0.189i)T \) |
| 7 | \( 1 + (-0.200 + 2.63i)T \) |
| 23 | \( 1 + (-4.03 - 2.59i)T \) |
| good | 2 | \( 1 + (1.49 + 1.42i)T + (0.0951 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-2.56 + 2.01i)T + (1.17 - 4.85i)T^{2} \) |
| 11 | \( 1 + (-2.38 + 2.27i)T + (0.523 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.65 - 5.81i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-6.09 + 3.13i)T + (9.86 - 13.8i)T^{2} \) |
| 19 | \( 1 + (4.63 + 2.39i)T + (11.0 + 15.4i)T^{2} \) |
| 29 | \( 1 + (-1.05 - 0.680i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.03 + 2.99i)T + (-24.3 + 19.1i)T^{2} \) |
| 37 | \( 1 + (5.16 - 2.06i)T + (26.7 - 25.5i)T^{2} \) |
| 41 | \( 1 + (-0.196 - 1.36i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (2.12 + 2.44i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.40 + 4.77i)T + (-17.3 + 50.0i)T^{2} \) |
| 59 | \( 1 + (-0.549 - 0.0525i)T + (57.9 + 11.1i)T^{2} \) |
| 61 | \( 1 + (5.94 + 1.14i)T + (56.6 + 22.6i)T^{2} \) |
| 67 | \( 1 + (2.06 - 8.52i)T + (-59.5 - 30.7i)T^{2} \) |
| 71 | \( 1 + (3.17 - 0.933i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-0.456 - 9.57i)T + (-72.6 + 6.93i)T^{2} \) |
| 79 | \( 1 + (-9.61 + 13.5i)T + (-25.8 - 74.6i)T^{2} \) |
| 83 | \( 1 + (0.164 - 1.14i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-2.61 + 7.55i)T + (-69.9 - 55.0i)T^{2} \) |
| 97 | \( 1 + (-2.47 - 17.1i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−10.59717268354924851927765395045, −9.726951242761787962709401653609, −9.191770007659170436238815944235, −8.469649910378396708547848069564, −7.00944348927297979082520091423, −5.99205364435694269806664036220, −4.85638122694506014184066964704, −3.56416915769363410845597155262, −1.71138185521706013112094788038, −0.947908099525024893189009641509,
1.54811637906161783218290920518, 3.20845985416915822708213433865, 5.37370839362172371784609008698, 6.08785749129413891887384438520, 6.51675275239269978903547047936, 7.70030165032863054263513596104, 8.549149187193183780780144423269, 9.495759871453424213191628313799, 10.32661117875046740412319740938, 10.73484654308673972432504604028
