L(s) = 1 | + (−2.15 + 0.578i)2-s + (−1.42 + 0.988i)3-s + (2.59 − 1.5i)4-s + (2.5 − 2.95i)6-s + (2.55 + 0.684i)7-s + (−1.58 + 1.58i)8-s + (1.04 − 2.81i)9-s + (−2.21 + 4.70i)12-s + (−3.74 − 3.74i)13-s − 5.91·14-s + (−0.500 + 0.866i)16-s + (−1.15 + 4.31i)17-s + (−0.631 + 6.67i)18-s + (−1.73 − i)19-s + (−4.31 + 1.55i)21-s + ⋯ |
L(s) = 1 | + (−1.52 + 0.409i)2-s + (−0.821 + 0.570i)3-s + (1.29 − 0.750i)4-s + (1.02 − 1.20i)6-s + (0.965 + 0.258i)7-s + (−0.559 + 0.559i)8-s + (0.348 − 0.937i)9-s + (−0.638 + 1.35i)12-s + (−1.03 − 1.03i)13-s − 1.58·14-s + (−0.125 + 0.216i)16-s + (−0.280 + 1.04i)17-s + (−0.148 + 1.57i)18-s + (−0.397 − 0.229i)19-s + (−0.940 + 0.338i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.240034 - 0.152490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.240034 - 0.152490i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.42 - 0.988i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.55 - 0.684i)T \) |
good | 2 | \( 1 + (2.15 - 0.578i)T + (1.73 - i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.74 + 3.74i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.15 - 4.31i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.73 + i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.73 + 6.47i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.73 + 10.2i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 5.91iT - 41T^{2} \) |
| 43 | \( 1 + (1.87 + 1.87i)T + 43iT^{2} \) |
| 47 | \( 1 + (-8.63 + 2.31i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-4.31 - 1.15i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (5.91 + 10.2i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.55 + 0.684i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + (1.36 - 5.11i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.73 + i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.58 - 1.58i)T - 83iT^{2} \) |
| 89 | \( 1 + (-2.95 + 5.12i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.48 - 7.48i)T - 97iT^{2} \) |
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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
−10.72174719906040749866041349737, −9.816420972778458681044930265917, −8.941670022067036152208303260291, −8.132530230680447142883108944622, −7.30543065169162663024776116354, −6.22734171153179807140764137279, −5.29574546013992422517292893765, −4.15567521230263079279991420342, −2.05075242819577907843712651699, −0.31940633012054008662272478736,
1.36490059584203482432231622161, 2.29437007544275936298546375319, 4.48878359903851607475367940742, 5.50919478010463426935222394106, 7.18506907784649189981954610804, 7.26293912233020515379532856788, 8.382549182859799367971000067785, 9.334529134289725993466901275545, 10.17021726311980723727216257812, 10.98760195441068092316744609959