| L(s) = 1 | + 675. i·5-s + 907.·7-s + 1.89e4i·11-s + 6.23e3·13-s − 1.51e4i·17-s + 1.81e5·19-s + 4.21e5i·23-s − 6.60e4·25-s − 1.90e5i·29-s − 1.98e5·31-s + 6.13e5i·35-s + 7.00e5·37-s − 6.28e5i·41-s + 5.91e6·43-s + 6.30e6i·47-s + ⋯ |
| L(s) = 1 | + 1.08i·5-s + 0.377·7-s + 1.29i·11-s + 0.218·13-s − 0.181i·17-s + 1.39·19-s + 1.50i·23-s − 0.169·25-s − 0.268i·29-s − 0.214·31-s + 0.408i·35-s + 0.373·37-s − 0.222i·41-s + 1.73·43-s + 1.29i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.180089361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180089361\) |
| \(L(5)\) |
|
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 | \( 1 - 907.T \) |
| good | 5 | \( 1 - 675. iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 1.89e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 6.23e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.51e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.81e5T + 1.69e10T^{2} \) |
| 23 | \( 1 - 4.21e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.90e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 1.98e5T + 8.52e11T^{2} \) |
| 37 | \( 1 - 7.00e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 6.28e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 5.91e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 6.30e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 2.32e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.34e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 9.18e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 6.29e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.43e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 4.04e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.00e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 5.01e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 4.17e6iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.41e8T + 7.83e15T^{2} \) |
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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
−11.00852863013389899177281950190, −9.968796188213892519784999794089, −9.256108981832834574870571420508, −7.63028058033056814847046852173, −7.26596594307981441348533434756, −6.00293660347802388332434721803, −4.84849658898857210118125465555, −3.57928426814005263404276585906, −2.48408038980534048788302465314, −1.28299033993337289642819445271,
0.52029765120857519943884476616, 1.27483880802214582947497655272, 2.85516997607308354120429081301, 4.15590845072097701677735088964, 5.20924529359381057486912227424, 6.06288413629868161587848293882, 7.49661809784812174016744292846, 8.551455294571511407086169928789, 9.013904268450212122640973938369, 10.33363052257672821797258732866
