| L(s) = 1 | + (−0.0681 − 0.474i)2-s + (0.415 − 0.909i)3-s + (1.69 − 0.498i)4-s + (0.654 − 0.755i)5-s + (−0.459 − 0.134i)6-s + (0.841 − 0.540i)7-s + (−0.750 − 1.64i)8-s + (−0.654 − 0.755i)9-s + (−0.402 − 0.258i)10-s + (−0.0556 + 0.386i)11-s + (0.251 − 1.75i)12-s + (0.211 + 0.136i)13-s + (−0.313 − 0.361i)14-s + (−0.415 − 0.909i)15-s + (2.25 − 1.44i)16-s + (−1.96 − 0.577i)17-s + ⋯ |
| L(s) = 1 | + (−0.0481 − 0.335i)2-s + (0.239 − 0.525i)3-s + (0.849 − 0.249i)4-s + (0.292 − 0.337i)5-s + (−0.187 − 0.0550i)6-s + (0.317 − 0.204i)7-s + (−0.265 − 0.580i)8-s + (−0.218 − 0.251i)9-s + (−0.127 − 0.0818i)10-s + (−0.0167 + 0.116i)11-s + (0.0727 − 0.505i)12-s + (0.0587 + 0.0377i)13-s + (−0.0838 − 0.0967i)14-s + (−0.107 − 0.234i)15-s + (0.562 − 0.361i)16-s + (−0.476 − 0.140i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0522 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0522 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.37922 - 1.30896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.37922 - 1.30896i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (0.936 - 4.70i)T \) |
| good | 2 | \( 1 + (0.0681 + 0.474i)T + (-1.91 + 0.563i)T^{2} \) |
| 5 | \( 1 + (-0.654 + 0.755i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (0.0556 - 0.386i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.211 - 0.136i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.96 + 0.577i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.48 + 0.435i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (3.13 + 0.920i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (0.777 + 1.70i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.05 - 4.67i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.76 + 3.19i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.13 + 2.49i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 2.48T + 47T^{2} \) |
| 53 | \( 1 + (1.56 - 1.00i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (3.66 + 2.35i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (1.11 + 2.44i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.115 - 0.801i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.715 - 4.97i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.62 - 1.35i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (-14.2 - 9.15i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-6.52 - 7.52i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.433 + 0.949i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.08 - 7.01i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−11.00384180012351939380917948720, −9.839535306093893858811379068439, −9.146492456315511847754281125479, −7.87975292785945496536043588774, −7.17606623235109605201332703014, −6.18592111323946256917839998646, −5.19908030456328556410790479377, −3.64150609496235121245226472832, −2.33738874902380452598497100633, −1.27698192122937639473520091696,
2.10965472771401862634969216461, 3.12114166737531741432630963203, 4.49113286268129002550276297401, 5.76231558996145652150420108519, 6.53444250623468704282676092853, 7.62008607378550804259624079551, 8.413185769715194244377193925825, 9.357088331737496956319938823016, 10.49717389838606292524219517983, 11.00906118313240334519381334191
