| L(s) = 1 | + (−1.55 − 0.771i)3-s + (−3.00 + 0.529i)5-s + (−0.251 + 0.435i)7-s + (1.80 + 2.39i)9-s + (2.58 − 1.49i)11-s + (0.364 + 1.00i)13-s + (5.06 + 1.49i)15-s + (0.866 + 1.03i)17-s + (1.71 − 4.00i)19-s + (0.725 − 0.480i)21-s + (8.83 + 1.55i)23-s + (4.03 − 1.46i)25-s + (−0.956 − 5.10i)27-s + (3.43 + 2.88i)29-s + (3.58 + 2.06i)31-s + ⋯ |
| L(s) = 1 | + (−0.895 − 0.445i)3-s + (−1.34 + 0.236i)5-s + (−0.0949 + 0.164i)7-s + (0.602 + 0.797i)9-s + (0.779 − 0.450i)11-s + (0.101 + 0.277i)13-s + (1.30 + 0.386i)15-s + (0.210 + 0.250i)17-s + (0.394 − 0.918i)19-s + (0.158 − 0.104i)21-s + (1.84 + 0.324i)23-s + (0.807 − 0.293i)25-s + (−0.184 − 0.982i)27-s + (0.638 + 0.535i)29-s + (0.643 + 0.371i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00473i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.836301 + 0.00197914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.836301 + 0.00197914i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.55 + 0.771i)T \) |
| 19 | \( 1 + (-1.71 + 4.00i)T \) |
| good | 5 | \( 1 + (3.00 - 0.529i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.251 - 0.435i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.58 + 1.49i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.364 - 1.00i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.866 - 1.03i)T + (-2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-8.83 - 1.55i)T + (21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.43 - 2.88i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.58 - 2.06i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 1.40iT - 37T^{2} \) |
| 41 | \( 1 + (4.19 + 1.52i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.539 + 3.05i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-1.56 + 1.86i)T + (-8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.97 - 11.2i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-8.56 + 7.18i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 8.62i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.94 - 8.27i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.02 - 11.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-9.45 - 3.44i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-0.310 + 0.852i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (6.19 + 3.57i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.7 + 5.00i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (9.45 + 11.2i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.29887192372025359421585834419, −10.55128452379296280497275726731, −9.165294448288408729929365089362, −8.238808435396833195091503011755, −7.13437339652507560375549851340, −6.68717947053761839681453768737, −5.34852321209995733980199166713, −4.32367546686602107048984966765, −3.10926711755812127458411218935, −0.985683325493321540298024457296,
0.862891176223548527454347426713, 3.41507469829268515362349280438, 4.28860684883810916441594175450, 5.16015123189303032533015332875, 6.47132738320226085840895835164, 7.29858289668807794525412604992, 8.314341288092900932130843844535, 9.398813771211572193975985553993, 10.29353934881577010252361287265, 11.22179629277665180478223575834
