L(s) = 1 | − 2·5-s + (−2.5 + 0.866i)7-s + 2·11-s + (1.5 − 2.59i)13-s + (−4 + 6.92i)17-s + (0.5 + 0.866i)19-s + 8·23-s − 25-s + (−2 − 3.46i)29-s + (−1.5 − 2.59i)31-s + (5 − 1.73i)35-s + (0.5 + 0.866i)37-s + (−3 + 5.19i)41-s + (−5.5 − 9.52i)43-s + (−3 + 5.19i)47-s + ⋯ |
L(s) = 1 | − 0.894·5-s + (−0.944 + 0.327i)7-s + 0.603·11-s + (0.416 − 0.720i)13-s + (−0.970 + 1.68i)17-s + (0.114 + 0.198i)19-s + 1.66·23-s − 0.200·25-s + (−0.371 − 0.643i)29-s + (−0.269 − 0.466i)31-s + (0.845 − 0.292i)35-s + (0.0821 + 0.142i)37-s + (−0.468 + 0.811i)41-s + (−0.838 − 1.45i)43-s + (−0.437 + 0.757i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7085218836\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7085218836\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.5 - 0.866i)T \) |
good | 5 | \( 1 + 2T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (4 - 6.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (2 + 3.46i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.5 + 2.59i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 - 0.866i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (5.5 + 9.52i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 + 5.19i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 + 11.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 + 1.73i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (5 + 8.66i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.707854946353923443239681425848, −8.194469277496348809008106029039, −7.24123618063053075935385709595, −6.45086571606035390087381523018, −5.86078088106469058537955594109, −4.70302044051655467839329026336, −3.70662862261236053323045136262, −3.28079800147793032970158999794, −1.86101504196763701920412081834, −0.28818875713798665331183373293,
1.05230229789107324205565502663, 2.68493372320211054448495936094, 3.51980597016098720127641130295, 4.29965849132331583558380523214, 5.12458567629727289600565990837, 6.33355362055912958846654969489, 7.16446439614433590326069326971, 7.24895345953278475290929837114, 8.821002278736665241454543459309, 8.973017302976633538447223661831