| L(s) = 1 | + (0.857 + 1.12i)2-s + (0.978 + 1.42i)3-s + (−0.528 + 1.92i)4-s + (0.00231 + 0.0175i)5-s + (−0.767 + 2.32i)6-s + (−0.191 − 0.714i)7-s + (−2.62 + 1.06i)8-s + (−1.08 + 2.79i)9-s + (−0.0177 + 0.0176i)10-s + (0.235 − 0.180i)11-s + (−3.27 + 1.13i)12-s + (0.822 − 1.07i)13-s + (0.639 − 0.827i)14-s + (−0.0228 + 0.0204i)15-s + (−3.44 − 2.03i)16-s + 3.82·17-s + ⋯ |
| L(s) = 1 | + (0.606 + 0.795i)2-s + (0.564 + 0.825i)3-s + (−0.264 + 0.964i)4-s + (0.00103 + 0.00785i)5-s + (−0.313 + 0.949i)6-s + (−0.0723 − 0.270i)7-s + (−0.927 + 0.374i)8-s + (−0.361 + 0.932i)9-s + (−0.00561 + 0.00558i)10-s + (0.0709 − 0.0544i)11-s + (−0.945 + 0.326i)12-s + (0.228 − 0.297i)13-s + (0.170 − 0.221i)14-s + (−0.00589 + 0.00528i)15-s + (−0.860 − 0.509i)16-s + 0.927·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.941792 + 1.68927i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.941792 + 1.68927i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.857 - 1.12i)T \) |
| 3 | \( 1 + (-0.978 - 1.42i)T \) |
| good | 5 | \( 1 + (-0.00231 - 0.0175i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.191 + 0.714i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.235 + 0.180i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-0.822 + 1.07i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 3.82T + 17T^{2} \) |
| 19 | \( 1 + (-2.78 + 1.15i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (4.24 + 1.13i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.0726 + 0.00956i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (1.05 - 0.606i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.37 + 8.15i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (2.68 - 10.0i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (0.251 + 0.327i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-3.79 - 2.19i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.729 + 1.76i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (11.5 - 1.51i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.53 + 11.6i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (4.33 - 5.64i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-3.05 - 3.05i)T + 71iT^{2} \) |
| 73 | \( 1 + (4.64 - 4.64i)T - 73iT^{2} \) |
| 79 | \( 1 + (-2.64 + 4.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.87 + 0.510i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-8.84 + 8.84i)T - 89iT^{2} \) |
| 97 | \( 1 + (-6.08 + 10.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.32810738226908559166420021297, −11.20260851072032753987234032386, −10.10483127112028145570098356425, −9.143904331201536517916677252170, −8.181986828030594498487380473381, −7.38312198959062202760421545856, −6.00761764842098318939387973024, −4.98768694034649900808108153259, −3.91587654211426819913186979292, −2.90491252906783517919323286652,
1.37474298786760535436679348971, 2.76894179422392828912214512587, 3.84229641666187661818714067822, 5.41684896109686857180370663544, 6.39714018345348820051240977280, 7.59763802187949645106504817948, 8.802141995989580132642048148053, 9.619237195057787507543958789785, 10.69175201599039975882762035354, 12.07663790455703130278954226945
