L(s) = 1 | − 4.38i·5-s + i·7-s − 2.31·11-s − 4.73·13-s − 5.27i·17-s + 5i·19-s − 0.517·23-s − 14.1·25-s + 4.24i·29-s + 3.53i·31-s + 4.38·35-s − 4.46·37-s + 6.45i·41-s + 1.26i·43-s + 8.10·47-s + ⋯ |
L(s) = 1 | − 1.95i·5-s + 0.377i·7-s − 0.696·11-s − 1.31·13-s − 1.28i·17-s + 1.14i·19-s − 0.107·23-s − 2.83·25-s + 0.787i·29-s + 0.635i·31-s + 0.740·35-s − 0.733·37-s + 1.00i·41-s + 0.193i·43-s + 1.18·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8015096804\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8015096804\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 4.38iT - 5T^{2} \) |
| 11 | \( 1 + 2.31T + 11T^{2} \) |
| 13 | \( 1 + 4.73T + 13T^{2} \) |
| 17 | \( 1 + 5.27iT - 17T^{2} \) |
| 19 | \( 1 - 5iT - 19T^{2} \) |
| 23 | \( 1 + 0.517T + 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 3.53iT - 31T^{2} \) |
| 37 | \( 1 + 4.46T + 37T^{2} \) |
| 41 | \( 1 - 6.45iT - 41T^{2} \) |
| 43 | \( 1 - 1.26iT - 43T^{2} \) |
| 47 | \( 1 - 8.10T + 47T^{2} \) |
| 53 | \( 1 - 5.93iT - 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 8.92T + 61T^{2} \) |
| 67 | \( 1 + 9.66iT - 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 0.517iT - 89T^{2} \) |
| 97 | \( 1 - 0.928T + 97T^{2} \) |
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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
−8.215692611129798149389617732983, −7.65305740332449679102422518616, −6.85251204334669031459795056318, −5.63864538844424297627866005233, −5.22834393307799241497571772510, −4.79570545187606676258925098186, −3.93113120071195189445366304811, −2.77126538799763323180015285737, −1.87873516534730473316562516984, −0.855021973796694489965699293567,
0.24150242015579209210053678799, 2.25268116770506600838995982866, 2.44800274358054743428837571586, 3.51617740596616161805759118630, 4.12305299684380724796569864508, 5.22258753322270790206905781927, 5.96067776048972145569267458985, 6.78892941122885752285652058517, 7.18229024209235176679797702754, 7.74674986162233918432572059656