L(s) = 1 | + (1.5 − 0.866i)3-s + 5·7-s + (1.5 − 2.59i)9-s + (−4.5 − 2.59i)13-s + (4 − 1.73i)19-s + (7.5 − 4.33i)21-s + (−2.5 + 4.33i)25-s − 5.19i·27-s + 8.66i·31-s + 5.19i·37-s − 9·39-s + (−6.5 − 11.2i)43-s + 18·49-s + (4.5 − 6.06i)57-s + (−0.5 + 0.866i)61-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + 1.88·7-s + (0.5 − 0.866i)9-s + (−1.24 − 0.720i)13-s + (0.917 − 0.397i)19-s + (1.63 − 0.944i)21-s + (−0.5 + 0.866i)25-s − 0.999i·27-s + 1.55i·31-s + 0.854i·37-s − 1.44·39-s + (−0.991 − 1.71i)43-s + 2.57·49-s + (0.596 − 0.802i)57-s + (−0.0640 + 0.110i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.33318 - 0.900197i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.33318 - 0.900197i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 19 | \( 1 + (-4 + 1.73i)T \) |
good | 5 | \( 1 + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 5T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (4.5 + 2.59i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 5.19iT - 37T^{2} \) |
| 41 | \( 1 + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (6.5 + 11.2i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.5 - 6.06i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (8.5 + 14.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (12 - 6.92i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.928612178167159912809993714179, −8.957020556546414183889186123776, −8.216998760458442233424620998517, −7.56938337420508142754658130787, −6.98042654099905501436006492195, −5.36620250397231844451617001840, −4.80394077382316967124084758876, −3.45352670381144064493442280730, −2.30585572546382674002330257810, −1.28526050084237823454170558749,
1.70088139618554623603763509828, 2.56707010043686208432243447665, 4.09801939102367821626202153730, 4.68743780324641005896945955857, 5.54965912198979869561048570285, 7.14651385057017684132040019813, 7.88813762640433061363143145341, 8.333419746413785022011167173882, 9.469820562591393852898259193574, 9.945658478946246537714761087508