| L(s) = 1 | − 42.6·3-s + (560. + 276. i)5-s + 869.·7-s − 4.74e3·9-s − 2.47e4i·11-s − 4.11e4i·13-s + (−2.39e4 − 1.18e4i)15-s − 4.09e4i·17-s − 7.95e4i·19-s − 3.71e4·21-s − 4.28e5·23-s + (2.37e5 + 3.10e5i)25-s + 4.82e5·27-s − 1.84e5·29-s + 1.54e5i·31-s + ⋯ |
| L(s) = 1 | − 0.526·3-s + (0.896 + 0.443i)5-s + 0.362·7-s − 0.722·9-s − 1.68i·11-s − 1.44i·13-s + (−0.472 − 0.233i)15-s − 0.489i·17-s − 0.610i·19-s − 0.190·21-s − 1.53·23-s + (0.607 + 0.794i)25-s + 0.907·27-s − 0.261·29-s + 0.167i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.1955295300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1955295300\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-560. - 276. i)T \) |
| good | 3 | \( 1 + 42.6T + 6.56e3T^{2} \) |
| 7 | \( 1 - 869.T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.47e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 4.11e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + 4.09e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 7.95e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 4.28e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 1.84e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.54e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 1.72e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 2.00e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 1.43e6T + 1.16e13T^{2} \) |
| 47 | \( 1 - 4.98e6T + 2.38e13T^{2} \) |
| 53 | \( 1 - 6.97e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 5.65e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 6.73e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 4.99e6T + 4.06e14T^{2} \) |
| 71 | \( 1 + 2.07e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.31e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 - 7.49e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 8.75e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 1.14e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 8.74e7iT - 7.83e15T^{2} \) |
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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
−9.890633491189682470891279834763, −8.692376688102952978352178348835, −7.945601001082983991641799525735, −6.49088030341950734174952372231, −5.76153380860528602796382451426, −5.16373312524453548208747878264, −3.35574700480986056127507876223, −2.56537053794254557573176414079, −1.02507799927047195284210388037, −0.04346591267769278752201478789,
1.65347140912347331071524542510, 2.13691329216519513089092887518, 4.07989456772909845144312608922, 4.95466010464726486855774631396, 5.92781806559801042815260941555, 6.75275414219058079140731343782, 7.991307933115099666644971487753, 9.080368306390125097886384616461, 9.831641304106521899644186861621, 10.68951668391603932083473117156
