L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − 1.56i·5-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.781 − 1.35i)10-s + (2.48 + 1.43i)11-s + (2.99 + 2.00i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.11 + 1.93i)17-s + (−6.26 + 3.61i)19-s + (1.35 − 0.781i)20-s + (1.43 + 2.48i)22-s + (0.833 − 1.44i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.699i·5-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.247 − 0.428i)10-s + (0.748 + 0.432i)11-s + (0.830 + 0.556i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.270 + 0.468i)17-s + (−1.43 + 0.830i)19-s + (0.302 − 0.174i)20-s + (0.305 + 0.529i)22-s + (0.173 − 0.301i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.483004035\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.483004035\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.866 - 0.5i)T \) |
| 13 | \( 1 + (-2.99 - 2.00i)T \) |
good | 5 | \( 1 + 1.56iT - 5T^{2} \) |
| 11 | \( 1 + (-2.48 - 1.43i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.11 - 1.93i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (6.26 - 3.61i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.833 + 1.44i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.41 + 4.18i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.597iT - 31T^{2} \) |
| 37 | \( 1 + (-0.0333 - 0.0192i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.88 - 3.97i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.04 - 8.73i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7.02iT - 47T^{2} \) |
| 53 | \( 1 - 5.98T + 53T^{2} \) |
| 59 | \( 1 + (0.776 - 0.448i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.12 - 12.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.42 + 0.820i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.98 + 1.14i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 - 4.26T + 79T^{2} \) |
| 83 | \( 1 - 4.94iT - 83T^{2} \) |
| 89 | \( 1 + (2.09 + 1.21i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.23 - 2.44i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.325682861885630036064654778026, −8.607544936891610342749944323932, −8.037356086098102184663701297798, −6.81394036175540004407261590616, −6.27608401239095868569491536960, −5.51042994829537604547785030778, −4.23085033681988316625214423131, −4.05969386808096150298022597000, −2.57009290462463113762655907093, −1.32438888429897324348160615643,
0.884122202246548482612109342596, 2.39991897578885041132813792532, 3.30308087923947761213260794989, 4.00563555074576647604197488362, 5.08985064098628792087465970329, 6.11770329685873681563835628279, 6.64082985295823651550907442276, 7.43497358279463259086057394714, 8.649430865346620060428415471582, 9.229709581906116679830668668932