L(s) = 1 | + (−0.707 + 0.707i)3-s + (1.27 + 1.27i)5-s + 0.158i·7-s − 1.00i·9-s + (3.79 + 3.79i)11-s + (−4.21 + 4.21i)13-s − 1.79·15-s + 3.05·17-s + (2.15 − 2.15i)19-s + (−0.112 − 0.112i)21-s − 2.82i·23-s − 1.76i·25-s + (0.707 + 0.707i)27-s + (2.09 − 2.09i)29-s − 4.15·31-s + ⋯ |
L(s) = 1 | + (−0.408 + 0.408i)3-s + (0.568 + 0.568i)5-s + 0.0600i·7-s − 0.333i·9-s + (1.14 + 1.14i)11-s + (−1.16 + 1.16i)13-s − 0.464·15-s + 0.740·17-s + (0.495 − 0.495i)19-s + (−0.0245 − 0.0245i)21-s − 0.589i·23-s − 0.353i·25-s + (0.136 + 0.136i)27-s + (0.389 − 0.389i)29-s − 0.746·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.988890 + 0.560387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.988890 + 0.560387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
good | 5 | \( 1 + (-1.27 - 1.27i)T + 5iT^{2} \) |
| 7 | \( 1 - 0.158iT - 7T^{2} \) |
| 11 | \( 1 + (-3.79 - 3.79i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.21 - 4.21i)T - 13iT^{2} \) |
| 17 | \( 1 - 3.05T + 17T^{2} \) |
| 19 | \( 1 + (-2.15 + 2.15i)T - 19iT^{2} \) |
| 23 | \( 1 + 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (-2.09 + 2.09i)T - 29iT^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 37 | \( 1 + (5.98 + 5.98i)T + 37iT^{2} \) |
| 41 | \( 1 + 2.60iT - 41T^{2} \) |
| 43 | \( 1 + (5.75 + 5.75i)T + 43iT^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + (-3.55 - 3.55i)T + 53iT^{2} \) |
| 59 | \( 1 + (4 + 4i)T + 59iT^{2} \) |
| 61 | \( 1 + (-3.66 + 3.66i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.767 - 0.767i)T - 67iT^{2} \) |
| 71 | \( 1 + 0.317iT - 71T^{2} \) |
| 73 | \( 1 + 1.33iT - 73T^{2} \) |
| 79 | \( 1 - 9.69T + 79T^{2} \) |
| 83 | \( 1 + (0.115 - 0.115i)T - 83iT^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 + 0.571T + 97T^{2} \) |
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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
−12.28394846118638302003926688258, −11.92860753412365095676112117674, −10.60063098243362821913589421191, −9.746443733095670553139433762766, −9.099073856968484862542039367852, −7.23679079770375296003894875878, −6.58402442750176547034362222669, −5.18640800857765897411880961294, −4.03125249694446395002958160784, −2.17333486573467991237480334359,
1.24264254162198050576480100326, 3.30929555236875743010133731146, 5.17148727733757249257006093125, 5.88520125345707706699712368694, 7.20898993143409779916004574814, 8.322741191113076581470058029429, 9.447744822486151092377390517691, 10.36889773308737704783543982604, 11.62323711111200958754092101823, 12.32427035096230560803436669269