L(s) = 1 | + (0.416 − 0.240i)5-s + (2.88 − 4.99i)11-s + 1.41i·13-s + (−19.9 − 11.5i)17-s + (−9.24 + 5.33i)19-s + (−3.29 − 5.71i)23-s + (−12.3 + 21.4i)25-s + 6.20·29-s + (36.3 + 20.9i)31-s + (30.0 + 52.0i)37-s + 48.8i·41-s − 51.5·43-s + (−16.6 + 9.62i)47-s + (41.0 − 71.1i)53-s − 2.77i·55-s + ⋯ |
L(s) = 1 | + (0.0832 − 0.0480i)5-s + (0.261 − 0.453i)11-s + 0.109i·13-s + (−1.17 − 0.678i)17-s + (−0.486 + 0.280i)19-s + (−0.143 − 0.248i)23-s + (−0.495 + 0.858i)25-s + 0.213·29-s + (1.17 + 0.676i)31-s + (0.812 + 1.40i)37-s + 1.19i·41-s − 1.19·43-s + (−0.354 + 0.204i)47-s + (0.774 − 1.34i)53-s − 0.0503i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.379 - 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9594179970\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9594179970\) |
\(L(2)\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + (-0.416 + 0.240i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-2.88 + 4.99i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 1.41iT - 169T^{2} \) |
| 17 | \( 1 + (19.9 + 11.5i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (9.24 - 5.33i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (3.29 + 5.71i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 6.20T + 841T^{2} \) |
| 31 | \( 1 + (-36.3 - 20.9i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-30.0 - 52.0i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 48.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 51.5T + 1.84e3T^{2} \) |
| 47 | \( 1 + (16.6 - 9.62i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-41.0 + 71.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (80.1 + 46.2i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (4.32 - 2.49i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-1.10 + 1.91i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 80.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (12.0 + 6.95i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-32.4 - 56.1i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 118. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (90.2 - 52.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 31.7iT - 9.40e3T^{2} \) |
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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
−9.423335046217490616602215262558, −8.470186957932562475971831803670, −7.993595566886028066342601333332, −6.67491152338627510174825050511, −6.44468119862027923252136803258, −5.18540244875523035864377657550, −4.48938701620222602961572232856, −3.42255923543732305447884947690, −2.42594691879907451591520146914, −1.20372187565671766066202313094,
0.25603272363798097412302059293, 1.80420375802518089175639871628, 2.66726070221074051700445180966, 4.03912510938754897263019618992, 4.53177019285262985692657281535, 5.77914506718513699611229181192, 6.42896857374877495148160282466, 7.24759852365167272195450663264, 8.151647719489264847552733622275, 8.850351952844977549175227007285