L(s) = 1 | + i·3-s + 1.41i·5-s − 2.82·7-s − 9-s + 4.24i·13-s − 1.41·15-s + 4·17-s + 8i·19-s − 2.82i·21-s + 5.65·23-s + 2.99·25-s − i·27-s + 1.41i·29-s − 2.82·31-s − 4.00i·35-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.632i·5-s − 1.06·7-s − 0.333·9-s + 1.17i·13-s − 0.365·15-s + 0.970·17-s + 1.83i·19-s − 0.617i·21-s + 1.17·23-s + 0.599·25-s − 0.192i·27-s + 0.262i·29-s − 0.508·31-s − 0.676i·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.010458778\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.010458778\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 4.24iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 8iT - 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 1.41iT - 29T^{2} \) |
| 31 | \( 1 + 2.82T + 31T^{2} \) |
| 37 | \( 1 + 4.24iT - 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 1.41iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−9.328476386135672262440786619000, −8.423191119022569354970572412736, −7.48104947475872013068134609657, −6.76660274605068201681933321881, −6.10430798894862212661469698803, −5.33509088936449280938034753261, −4.24995216240937495066672679690, −3.46118175310632885715589066272, −2.90531902820461200933785007062, −1.53306298593563105332752533346,
0.33966314377470585649426705078, 1.24498295179402114269579511218, 2.88232902276880888445976588399, 3.13965950913570075205202074944, 4.58470776458149586910119616845, 5.32013589909200682516603994779, 6.06231450765852555130522116094, 6.94394229867907381491437696779, 7.44405634828031689220123986130, 8.434773443276710275112361327515