| L(s) = 1 | + (2.57 + 0.689i)2-s + (0.974 + 5.10i)3-s + (−0.789 − 0.455i)4-s + (−1.01 + 13.7i)6-s + (0.400 − 1.49i)7-s + (−16.7 − 16.7i)8-s + (−25.1 + 9.94i)9-s + (−61.6 + 35.5i)11-s + (1.55 − 4.47i)12-s + (10.3 + 38.6i)13-s + (2.05 − 3.56i)14-s + (−27.9 − 48.3i)16-s + (−45.8 + 45.8i)17-s + (−71.4 + 8.28i)18-s + 3.05i·19-s + ⋯ |
| L(s) = 1 | + (0.909 + 0.243i)2-s + (0.187 + 0.982i)3-s + (−0.0986 − 0.0569i)4-s + (−0.0687 + 0.938i)6-s + (0.0216 − 0.0806i)7-s + (−0.741 − 0.741i)8-s + (−0.929 + 0.368i)9-s + (−1.69 + 0.975i)11-s + (0.0374 − 0.107i)12-s + (0.220 + 0.824i)13-s + (0.0392 − 0.0680i)14-s + (−0.436 − 0.756i)16-s + (−0.654 + 0.654i)17-s + (−0.935 + 0.108i)18-s + 0.0368i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00170i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.000997618 + 1.16986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000997618 + 1.16986i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.974 - 5.10i)T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (-2.57 - 0.689i)T + (6.92 + 4i)T^{2} \) |
| 7 | \( 1 + (-0.400 + 1.49i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (61.6 - 35.5i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-10.3 - 38.6i)T + (-1.90e3 + 1.09e3i)T^{2} \) |
| 17 | \( 1 + (45.8 - 45.8i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 - 3.05iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (-140. + 37.6i)T + (1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (91.6 + 158. i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (89.3 - 154. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-26.4 - 26.4i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 + (-79.5 - 45.9i)T + (3.44e4 + 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-48.4 - 12.9i)T + (6.88e4 + 3.97e4i)T^{2} \) |
| 47 | \( 1 + (143. + 38.3i)T + (8.99e4 + 5.19e4i)T^{2} \) |
| 53 | \( 1 + (-182. - 182. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 + (-128. + 223. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-152. - 263. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-231. + 62.0i)T + (2.60e5 - 1.50e5i)T^{2} \) |
| 71 | \( 1 - 470. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (445. - 445. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (-134. + 77.7i)T + (2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (216. - 806. i)T + (-4.95e5 - 2.85e5i)T^{2} \) |
| 89 | \( 1 + 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-77.0 + 287. i)T + (-7.90e5 - 4.56e5i)T^{2} \) |
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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
−12.61100636018308072612209485692, −11.22756037838343210995765294696, −10.32740364341412225546677899241, −9.458807871165603750016537289223, −8.480788602702228586587880946644, −7.06344060927265374345948309757, −5.69045519137974404413585402029, −4.81162187184842148696037372592, −4.04050866673646333995397468862, −2.60581755470852127678109489257,
0.33007942848235065012797270475, 2.53024603609456868091443256185, 3.33490819698222648596189282418, 5.16607224271183021763872114500, 5.78279195853670797423923882716, 7.29501539512998580581532100109, 8.254613175525747449375550195195, 9.058627417712069955311235632004, 10.82328152762865049191794453247, 11.47485885558241445302481922578
